Theorem bren2 3293

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 3289	. . 3 |- (A ~~ B -> A ~<_ B)
2		sdomnen 3291	. . . 4 |- (A ~< B -> -. A ~~ B)
3	2	con2i 89	. . 3 |- (A ~~ B -> -. A ~< B)
4	1, 3	jca 236	. 2 |- (A ~~ B -> (A ~<_ B /\ -. A ~< B))
5		brdom2 3292	. . . 4 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
6	5	biimp 133	. . 3 |- (A ~<_ B -> (A ~< B \/ A ~~ B))
7	6	orcanai 515	. 2 |- ((A ~<_ B /\ -. A ~< B) -> A ~~ B)
8	4, 7	impbi 139	1 |- (A ~~ B <-> (A ~<_ B /\ -. A ~< B))

Syntax hints:  -. wn 1   <-> wb 127   \/ wo 195   /\ wa 196   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  alephsuc3 4955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-f1o 2437  df-en 3274  df-dom 3275  df-sdom 3276

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