Theorem bren2 7449

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 7445	. . 3 |- ( A ~~ B -> A ~<_ B )
2		sdomnen 7447	. . . 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 7448	. . . 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 4399   ~~ cen 7416   ~<_ cdom 7417   ~< csdm 7418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-f1o 5532  df-en 7420  df-dom 7421  df-sdom 7422

This theorem is referenced by:  marypha1lem  7793  tskwe  8230  infxpenlem  8290  cdainflem  8470  axcclem  8736  alephsuc3  8854  gchen1  8902  gchen2  8903  inatsk  9055  ufilen  19634  dirith2  22909  mblfinlem1  28575