Theorem bren2 4395

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 4391	. . 3 |- (A ~~ B -> A ~<_ B)
2		sdomnen 4393	. . . 4 |- (A ~< B -> -. A ~~ B)
3	2	con2i 97	. . 3 |- (A ~~ B -> -. A ~< B)
4	1, 3	jca 288	. 2 |- (A ~~ B -> (A ~<_ B /\ -. A ~< B))
5		brdom2 4394	. . . 4 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
6	5	biimp 151	. . 3 |- (A ~<_ B -> (A ~< B \/ A ~~ B))
7	6	orcanai 692	. 2 |- ((A ~<_ B /\ -. A ~< B) -> A ~~ B)
8	4, 7	impbi 157	1 |- (A ~~ B <-> (A ~<_ B /\ -. A ~< B))

Syntax hints:  -. wn 2   <-> wb 146   \/ wo 222   /\ wa 223   class class class wbr 2624   ~~ cen 4370   ~<_ cdom 4371   ~< csdm 4372

This theorem is referenced by:  alephsuc3 7587

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-f1o 3203  df-en 4374  df-dom 4375  df-sdom 4376

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