MHB Basic Inequality Prove: $A\leq B\wedge B\leq A \Rightarrow A=B$

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The discussion focuses on proving the statement that if A is less than or equal to B and B is less than or equal to A, then A must equal B. It references the trichotomy law, which asserts that for any two numbers A and B, one of three conditions must hold: A is greater than B, A is less than B, or A equals B. The participants clarify that since A is less than or equal to B, A cannot be greater than B, and since B is less than or equal to A, A cannot be less than B. The equivalence of the trichotomy law to the expression A < B or B < A or A = B is also questioned. The discussion emphasizes the logical structure underpinning the proof of equality based on the given inequalities.
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Prove:

$A\leq B\wedge B\leq A\Rightarrow A=B$
 
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Trichotomy: Given numbers A and B, one and only one must be true:
1) A> B
2) A< B
3) A= B

Since [math]A\le B[/math], A> B is not true.
Since [math]B\le A[/math], A< B is not true.
 
Is the trichotomy law you are using Equivelant to the following:
$A<B\vee B<A\vee A=B$
 
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