Theorem xrlenlt

Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by NM, 14-Oct-2005)

	Ref	Expression
Assertion	xrlenlt	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-br	$$⊢A\le B\leftrightarrow ⟨A,B⟩\in \le$$
2		opelxpi	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to ⟨A,B⟩\in \left({ℝ}^{*}\times {ℝ}^{*}\right)$$
3		df-le	$$⊢\le =\left({ℝ}^{*}\times {ℝ}^{*}\right)\setminus {<}^{-1}$$
4	3	eleq2i	$$⊢⟨A,B⟩\in \le \leftrightarrow ⟨A,B⟩\in \left(\left({ℝ}^{*}\times {ℝ}^{*}\right)\setminus {<}^{-1}\right)$$
5		eldif	$$⊢⟨A,B⟩\in \left(\left({ℝ}^{*}\times {ℝ}^{*}\right)\setminus {<}^{-1}\right)\leftrightarrow \left(⟨A,B⟩\in \left({ℝ}^{*}\times {ℝ}^{*}\right)\wedge \neg ⟨A,B⟩\in {<}^{-1}\right)$$
6	4 5	bitri	$$⊢⟨A,B⟩\in \le \leftrightarrow \left(⟨A,B⟩\in \left({ℝ}^{*}\times {ℝ}^{*}\right)\wedge \neg ⟨A,B⟩\in {<}^{-1}\right)$$
7	6	baib	$$⊢⟨A,B⟩\in \left({ℝ}^{*}\times {ℝ}^{*}\right)\to \left(⟨A,B⟩\in \le \leftrightarrow \neg ⟨A,B⟩\in {<}^{-1}\right)$$
8	2 7	syl	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(⟨A,B⟩\in \le \leftrightarrow \neg ⟨A,B⟩\in {<}^{-1}\right)$$
9	1 8	syl5bb	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg ⟨A,B⟩\in {<}^{-1}\right)$$
10		opelcnvg	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(⟨A,B⟩\in {<}^{-1}\leftrightarrow ⟨B,A⟩\in <\right)$$
11		df-br	$$⊢B<A\leftrightarrow ⟨B,A⟩\in <$$
12	10 11	syl6rbbr	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(B<A\leftrightarrow ⟨A,B⟩\in {<}^{-1}\right)$$
13	12	notbid	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(\neg B<A\leftrightarrow \neg ⟨A,B⟩\in {<}^{-1}\right)$$
14	9 13	bitr4d	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(A\le B\leftrightarrow \neg B<A\right)$$