Theorem gt-lt 47142

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

Assertion

Ref	Expression
gt-lt	⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴))

Proof of Theorem gt-lt

Step	Hyp	Ref	Expression
1		df-gt 47140	. . 3 ⊢ > = ◡ <
2	1	breqi 5111	. 2 ⊢ (𝐴 > 𝐵 ↔ 𝐴◡ < 𝐵)
3		brcnvg 5835	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴))
4	2, 3	bitrid 282	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵 ↔ 𝐵 < 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  Vcvv 3445   class class class wbr 5105  ^◡ccnv 5632   < clt 11188   > cgt 47138

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-cnv 5641  df-gt 47140

This theorem is referenced by: (None)