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Theorem domen 6875
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.
StepHypRef Expression
1 bren.1 . . 3  |-  B  e. 
_V
21brdom 6874 . 2  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 vex 2791 . . . . . 6  |-  f  e. 
_V
43f11o 5506 . . . . 5  |-  ( f : A -1-1-> B  <->  E. x
( f : A -1-1-onto-> x  /\  x  C_  B ) )
54exbii 1569 . . . 4  |-  ( E. f  f : A -1-1-> B  <->  E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B ) )
6 excom 1786 . . . 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 240 . . 3  |-  ( E. f  f : A -1-1-> B  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
8 bren 6871 . . . . . 6  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
98anbi1i 676 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
10 19.41v 1842 . . . . 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 243 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
1211exbii 1569 . . 3  |-  ( E. x ( A  ~~  x  /\  x  C_  B
)  <->  E. x E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
137, 12bitr4i 243 . 2  |-  ( E. f  f : A -1-1-> B  <->  E. x ( A  ~~  x  /\  x  C_  B
) )
142, 13bitri 240 1  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   -1-1->wf1 5252   -1-1-onto->wf1o 5254    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  domeng  6876  infcntss  7130  cdainf  7818  ramub2  13061  ram0  13069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-en 6864  df-dom 6865
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