Theorem xrletrid

Description: Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrletrid.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xrletrid.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	xrletrid.3	$$⊢\phi \to A\le B$$
	xrletrid.4	$$⊢\phi \to B\le A$$
Assertion	xrletrid	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		xrletrid.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrletrid.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrletrid.3	$$⊢\phi \to A\le B$$
4		xrletrid.4	$$⊢\phi \to B\le A$$
5		xrletri3	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A=B\leftrightarrow \left(A\le B\wedge B\le A\right)\right)$$
6	1 2 5	syl2anc	$$⊢\phi \to \left(A=B\leftrightarrow \left(A\le B\wedge B\le A\right)\right)$$
7	3 4 6	mpbir2and	$$⊢\phi \to A=B$$