Theorem bren2 7332

Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
bren2	|- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )

Proof of Theorem bren2

Step	Hyp	Ref	Expression
1		endom 7328	. . 3 |- ( A ~~ B -> A ~<_ B )
2		sdomnen 7330	. . . 4 |- ( A ~< B -> -. A ~~ B )
3	2	con2i 120	. . 3 |- ( A ~~ B -> -. A ~< B )
4	1, 3	jca 532	. 2 |- ( A ~~ B -> ( A ~<_ B /\ -. A ~< B ) )
5		brdom2 7331	. . . 4 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
6	5	biimpi 194	. . 3 |- ( A ~<_ B -> ( A ~< B \/ A ~~ B ) )
7	6	orcanai 904	. 2 |- ( ( A ~<_ B /\ -. A ~< B ) -> A ~~ B )
8	4, 7	impbii 188	1 |- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )

Syntax hints:   -. wn 3   <-> wb 184   \/ wo 368   /\ wa 369   class class class wbr 4287   ~~ cen 7299   ~<_ cdom 7300   ~< csdm 7301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-f1o 5420  df-en 7303  df-dom 7304  df-sdom 7305

This theorem is referenced by:  marypha1lem  7675  tskwe  8112  infxpenlem  8172  cdainflem  8352  axcclem  8618  alephsuc3  8736  gchen1  8784  gchen2  8785  inatsk  8937  ufilen  19483  dirith2  22757  mblfinlem1  28399