Theorem domen 6769

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 6768	. 2 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		vex 2755	. . . . . 6 |- f e. _V
4	3	f11o 5509	. . . . 5 |- ( f : A -1-1-> B <-> E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
5	4	exbii 1616	. . . 4 |- ( E. f f : A -1-1-> B <-> E. f E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
6		excom 1675	. . . 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 6765	. . . . . 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 1914	. . . . 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 1616	. . 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 1503   e. wcel 2160   _Vcvv 2752   C_ wss 3144   class class class wbr 4018   -1-1->wf1 5228   -1-1-onto->wf1o 5230   ~~ cen 6756   ~<_ cdom 6757

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-dm 4651  df-rn 4652  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-en 6759  df-dom 6760

This theorem is referenced by:  domeng  6770  php5dom  6881