Theorem domen 6990

Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)

Hypothesis

Ref	Expression
bren.1	|- B e. _V

Assertion

Ref	Expression
domen	|- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem domen

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren.1	. . 3 |- B e. _V
2	1	brdom 6989	. 2 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		vex 2818	. . . . . 6 |- f e. _V
4	3	f11o 5650	. . . . 5 |- ( f : A -1-1-> B <-> E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
5	4	exbii 1654	. . . 4 |- ( E. f f : A -1-1-> B <-> E. f E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
6		excom 1712	. . . 4 |- ( E. f E. x ( f : A -1-1-onto-> x /\ x C_ B ) <-> E. x E. f ( f : A -1-1-onto-> x /\ x C_ B ) )
7	5, 6	bitri 184	. . 3 |- ( E. f f : A -1-1-> B <-> E. x E. f ( f : A -1-1-onto-> x /\ x C_ B ) )
8		bren 6985	. . . . . 6 |- ( A ~~ x <-> E. f f : A -1-1-onto-> x )
9	8	anbi1i 458	. . . . 5 |- ( ( A ~~ x /\ x C_ B ) <-> ( E. f f : A -1-1-onto-> x /\ x C_ B ) )
10		19.41v 1954	. . . . 5 |- ( E. f ( f : A -1-1-onto-> x /\ x C_ B ) <-> ( E. f f : A -1-1-onto-> x /\ x C_ B ) )
11	9, 10	bitr4i 187	. . . 4 |- ( ( A ~~ x /\ x C_ B ) <-> E. f ( f : A -1-1-onto-> x /\ x C_ B ) )
12	11	exbii 1654	. . 3 |- ( E. x ( A ~~ x /\ x C_ B ) <-> E. x E. f ( f : A -1-1-onto-> x /\ x C_ B ) )
13	7, 12	bitr4i 187	. 2 |- ( E. f f : A -1-1-> B <-> E. x ( A ~~ x /\ x C_ B ) )
14	2, 13	bitri 184	1 |- ( A ~<_ B <-> E. x ( A ~~ x /\ x C_ B ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1541   e. wcel 2205   _Vcvv 2815   C_ wss 3213   class class class wbr 4111   -1-1->wf1 5351   -1-1-onto->wf1o 5353   ~~ cen 6975   ~<_ cdom 6976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-cnv 4759  df-dm 4761  df-rn 4762  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-en 6978  df-dom 6979

This theorem is referenced by:  domeng  6991  php5dom  7119