Theorem bren2 7618

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 7614	. . 3 |- ( A ~~ B -> A ~<_ B )
2		sdomnen 7616	. . . 4 |- ( A ~< B -> -. A ~~ B )
3	2	con2i 124	. . 3 |- ( A ~~ B -> -. A ~< B )
4	1, 3	jca 541	. 2 |- ( A ~~ B -> ( A ~<_ B /\ -. A ~< B ) )
5		brdom2 7617	. . . 4 |- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
6	5	biimpi 199	. . 3 |- ( A ~<_ B -> ( A ~< B \/ A ~~ B ) )
7	6	orcanai 927	. 2 |- ( ( A ~<_ B /\ -. A ~< B ) -> A ~~ B )
8	4, 7	impbii 192	1 |- ( A ~~ B <-> ( A ~<_ B /\ -. A ~< B ) )

Syntax hints:   -. wn 3   <-> wb 189   \/ wo 375   /\ wa 376   class class class wbr 4395   ~~ cen 7584   ~<_ cdom 7585   ~< csdm 7586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-f1o 5596  df-en 7588  df-dom 7589  df-sdom 7590

This theorem is referenced by:  marypha1lem  7965  tskwe  8402  infxpenlem  8462  cdainflem  8639  axcclem  8905  alephsuc3  9023  gchen1  9068  gchen2  9069  inatsk  9221  ufilen  21023  dirith2  24445  f1ocnt  28451  mblfinlem1  32041