Theorem gt-lt 46427

Description: Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
gt-lt	⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴))

Proof of Theorem gt-lt

Step	Hyp	Ref	Expression
1		df-gt 46425	. . 3 ⊢ > = ◡ <
2	1	breqi 5080	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		brcnvg 5788	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴))
4	2, 3	syl5bb 283	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  ^◡ccnv 5588   < clt 11009   > cgt 46423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-gt 46425

This theorem is referenced by: (None)