Theorem gte-lteh 48027

Description: Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

Hypotheses

Ref	Expression
gte-lteh.1	⊢ 𝐴 ∈ V
gte-lteh.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
gte-lteh	⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)

Proof of Theorem gte-lteh

Step	Hyp	Ref	Expression
1		df-gte 48023	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5147	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		gte-lteh.1	. . 3 ⊢ 𝐴 ∈ V
4		gte-lteh.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5875	. 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴)
6	2, 5	bitri 275	1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  ^◡ccnv 5668   ≤ cle 11250   ≥ cge-real 48021

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

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 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cnv 5677  df-gte 48023

This theorem is referenced by:  ex-gte  48030