Theorem gte-lteh 42467

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 e. _V
gte-lteh.2	|- B e. _V

Assertion

Ref	Expression
gte-lteh	|- ( A >_ B <-> B <_ A )

Proof of Theorem gte-lteh

Step	Hyp	Ref	Expression
1		df-gte 42463	. . 3 |- >_ = `' <_
2	1	breqi 4659	. 2 |- ( A >_ B <-> A `' <_ B )
3		gte-lteh.1	. . 3 |- A e. _V
4		gte-lteh.2	. . 3 |- B e. _V
5	3, 4	brcnv 5305	. 2 |- ( A `' <_ B <-> B <_ A )
6	2, 5	bitri 264	1 |- ( A >_ B <-> B <_ A )

Syntax hints:   <-> wb 196   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   <_ cle 10075   >_ cge-real 42461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-gte 42463

This theorem is referenced by:  ex-gte  42470