Theorem gt-lth 50115

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

Hypotheses

Ref	Expression
gt-lth.1	⊢ 𝐴 ∈ V
gt-lth.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
gt-lth	⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

Proof of Theorem gt-lth

Step	Hyp	Ref	Expression
1		df-gt 50111	. . 3 ⊢ > = ◡ <
2	1	breqi 5106	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		gt-lth.1	. . 3 ⊢ 𝐴 ∈ V
4		gt-lth.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5841	. 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)
6	2, 5	bitri 275	1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100  ^◡ccnv 5633   < clt 11180   > cgt 50109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5642  df-gt 50111

This theorem is referenced by:  ex-gt  50116