Theorem gt-lth 49720

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 49716	. . 3 ⊢ > = ◡ <
2	1	breqi 5116	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		gt-lth.1	. . 3 ⊢ 𝐴 ∈ V
4		gt-lth.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5849	. 2 ⊢ (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)
6	2, 5	bitri 275	1 ⊢ (𝐴 > 𝐵 ↔ 𝐵 < 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  Vcvv 3450   class class class wbr 5110  ^◡ccnv 5640   < clt 11215   > cgt 49714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-cnv 5649  df-gt 49716

This theorem is referenced by:  ex-gt  49721