Theorem gte-lteh 49245

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 49241	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5149	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		gte-lteh.1	. . 3 ⊢ 𝐴 ∈ V
4		gte-lteh.2	. . 3 ⊢ 𝐵 ∈ V
5	3, 4	brcnv 5893	. 2 ⊢ (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴)
6	2, 5	bitri 275	1 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  ^◡ccnv 5684   ≤ cle 11296   ≥ cge-real 49239

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-gte 49241

This theorem is referenced by:  ex-gte  49248