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Theorem domen 6870
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
Dummy variable  f is distinct from all other variables.

Proof of Theorem domen
StepHypRef Expression
1 bren.1 . . 3  |-  B  e. 
_V
21brdom 6869 . 2  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 vex 2792 . . . . . 6  |-  f  e. 
_V
43f11o 5471 . . . . 5  |-  ( f : A -1-1-> B  <->  E. x
( f : A -1-1-onto-> x  /\  x  C_  B ) )
54exbii 1570 . . . 4  |-  ( E. f  f : A -1-1-> B  <->  E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B ) )
6 excom 1787 . . . 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 ) )
75, 6bitri 242 . . 3  |-  ( E. f  f : A -1-1-> B  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
8 bren 6866 . . . . . 6  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
98anbi1i 678 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
10 19.41v 1843 . . . . 5  |-  ( E. f ( f : A -1-1-onto-> x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
119, 10bitr4i 245 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
1211exbii 1570 . . 3  |-  ( E. x ( A  ~~  x  /\  x  C_  B
)  <->  E. x E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
137, 12bitr4i 245 . 2  |-  ( E. f  f : A -1-1-> B  <->  E. x ( A  ~~  x  /\  x  C_  B
) )
142, 13bitri 242 1  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529    e. wcel 1685   _Vcvv 2789    C_ wss 3153   class class class wbr 4024   -1-1->wf1 5218   -1-1-onto->wf1o 5220    ~~ cen 6855    ~<_ cdom 6856
This theorem is referenced by:  domeng  6871  infcntss  7125  cdainf  7813  ramub2  13055  ram0  13063
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6859  df-dom 6860
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