Theorem gt-lth 51098

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 51094	. . 3 ⊢ > = ◡ <
2	1	breqi 5112	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		gt-lth.1	. . 3 ⊢ 𝐴 ∈ V
4		gt-lth.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5866	. 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)
6	2, 5	bitri 278	1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  ^◡ccnv 5654   < clt 11367   > cgt 51092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-gt 51094

This theorem is used by:  ex-gt  51099