Theorem gt-lth 48020

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 48016	. . 3 ⊢ > = ◡ <
2	1	breqi 5145	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		gt-lth.1	. . 3 ⊢ 𝐴 ∈ V
4		gt-lth.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5873	. 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)
6	2, 5	bitri 275	1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  Vcvv 3466   class class class wbr 5139  ^◡ccnv 5666   < clt 11247   > cgt 48014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-gt 48016

This theorem is referenced by:  ex-gt  48021