Theorem domen 6805

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 6804	. 2 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		vex 2763	. . . . . 6 |- f e. _V
4	3	f11o 5533	. . . . 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 6801	. . . . . 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 2164   _Vcvv 2760   C_ wss 3153   class class class wbr 4029   -1-1->wf1 5251   -1-1-onto->wf1o 5253   ~~ cen 6792   ~<_ cdom 6793

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-en 6795  df-dom 6796

This theorem is referenced by:  domeng  6806  php5dom  6919