Theorem gte-lteh 50453

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 50449	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5120	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		gte-lteh.1	. . 3 ⊢ 𝐴 ∈ V
4		gte-lteh.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5872	. 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴)
6	2, 5	bitri 278	1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)

Syntax hints:   ↔ wb 209   ∈ wcel 2150  Vcvv 3462   class class class wbr 5114  ^◡ccnv 5664   ≤ cle 11247   ≥ cge-real 50447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5673  df-gte 50449

This theorem is referenced by:  ex-gte  50456