Theorem gte-lteh 50230

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 50226	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5081	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		gte-lteh.1	. . 3 ⊢ 𝐴 ∈ V
4		gte-lteh.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5827	. 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴)
6	2, 5	bitri 277	1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2121  Vcvv 3433   class class class wbr 5075  ^◡ccnv 5620   ≤ cle 11175   ≥ cge-real 50224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-cnv 5629  df-gte 50226

This theorem is referenced by:  ex-gte  50233