Theorem domen 6840

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 6839	. 2 |- ( A ~<_ B <-> E. f f : A -1-1-> B )
3		vex 2775	. . . . . 6 |- f e. _V
4	3	f11o 5555	. . . . 5 |- ( f : A -1-1-> B <-> E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
5	4	exbii 1628	. . . 4 |- ( E. f f : A -1-1-> B <-> E. f E. x ( f : A -1-1-onto-> x /\ x C_ B ) )
6		excom 1687	. . . 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 6835	. . . . . 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 1926	. . . . 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 1628	. . 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 1515   e. wcel 2176   _Vcvv 2772   C_ wss 3166   class class class wbr 4044   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ~~ cen 6825   ~<_ cdom 6826

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-dm 4685  df-rn 4686  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6828  df-dom 6829

This theorem is referenced by:  domeng  6841  php5dom  6960