Theorem bren2 4530

Description: Equinumerosity expressed in terms of dominance and strict dominance.

Assertion

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

Proof of Theorem bren2

Step	Hyp	Ref	Expression
1		endom 4526	. . 3 |- (A ~~ B -> A ~<_ B)
2		sdomnen 4528	. . . 4 |- (A ~< B -> -. A ~~ B)
3	2	con2i 97	. . 3 |- (A ~~ B -> -. A ~< B)
4	1, 3	jca 286	. 2 |- (A ~~ B -> (A ~<_ B /\ -. A ~< B))
5		brdom2 4529	. . . 4 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
6	5	biimpi 149	. . 3 |- (A ~<_ B -> (A ~< B \/ A ~~ B))
7	6	orcanai 694	. 2 |- ((A ~<_ B /\ -. A ~< B) -> A ~~ B)
8	4, 7	impbii 155	1 |- (A ~~ B <-> (A ~<_ B /\ -. A ~< B))

Syntax hints:  -. wn 2   <-> wb 144   \/ wo 220   /\ wa 221   class class class wbr 2692   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507

This theorem is referenced by:  alephsuc3 7797  infenomsub 11450

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-br 2693  df-opab 2741  df-xp 3265  df-rel 3266  df-f1o 3278  df-en 4509  df-dom 4510  df-sdom 4511

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