Theorem bren2 7539

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 7535	. . 3 |- ( A ~~ B -> A ~<_ B )
2		sdomnen 7537	. . . 4 |- ( A ~< B -> -. A ~~ B )
3	2	con2i 120	. . 3 |- ( A ~~ B -> -. A ~< B )
4	1, 3	jca 530	. 2 |- ( A ~~ B -> ( A ~<_ B /\ -. A ~< B ) )
5		brdom2 7538	. . . 4 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
6	5	biimpi 194	. . 3 |- ( A ~<_ B -> ( A ~< B \/ A ~~ B ) )
7	6	orcanai 911	. 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 366   /\ wa 367   class class class wbr 4439   ~~ cen 7506   ~<_ cdom 7507   ~< csdm 7508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-f1o 5577  df-en 7510  df-dom 7511  df-sdom 7512

This theorem is referenced by:  marypha1lem  7885  tskwe  8322  infxpenlem  8382  cdainflem  8562  axcclem  8828  alephsuc3  8946  gchen1  8992  gchen2  8993  inatsk  9145  ufilen  20597  dirith2  23911  mblfinlem1  30291