Theorem gt-lth 46100

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 46096	. . 3 ⊢ > = ◡ <
2	1	breqi 5059	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		gt-lth.1	. . 3 ⊢ 𝐴 ∈ V
4		gt-lth.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5751	. 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)
6	2, 5	bitri 278	1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

Syntax hints:   ↔ wb 209   ∈ wcel 2110  Vcvv 3408   class class class wbr 5053  ^◡ccnv 5550   < clt 10867   > cgt 46094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-cnv 5559  df-gt 46096

This theorem is referenced by:  ex-gt  46101