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Theorem domen 6298
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 6297 . 2  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 vex 2605 . . . . . 6  |-  f  e. 
_V
43f11o 5190 . . . . 5  |-  ( f : A -1-1-> B  <->  E. x
( f : A -1-1-onto-> x  /\  x  C_  B ) )
54exbii 1537 . . . 4  |-  ( E. f  f : A -1-1-> B  <->  E. f E. x ( f : A -1-1-onto-> x  /\  x  C_  B ) )
6 excom 1595 . . . 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 182 . . 3  |-  ( E. f  f : A -1-1-> B  <->  E. x E. f ( f : A -1-1-onto-> x  /\  x  C_  B ) )
8 bren 6294 . . . . . 6  |-  ( A 
~~  x  <->  E. f 
f : A -1-1-onto-> x )
98anbi1i 446 . . . . 5  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  ( E. f  f : A -1-1-onto-> x  /\  x  C_  B ) )
10 19.41v 1824 . . . . 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 185 . . . 4  |-  ( ( A  ~~  x  /\  x  C_  B )  <->  E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
1211exbii 1537 . . 3  |-  ( E. x ( A  ~~  x  /\  x  C_  B
)  <->  E. x E. f
( f : A -1-1-onto-> x  /\  x  C_  B ) )
137, 12bitr4i 185 . 2  |-  ( E. f  f : A -1-1-> B  <->  E. x ( A  ~~  x  /\  x  C_  B
) )
142, 13bitri 182 1  |-  ( A  ~<_  B  <->  E. x ( A 
~~  x  /\  x  C_  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2602    C_ wss 2974   class class class wbr 3793   -1-1->wf1 4929   -1-1-onto->wf1o 4931    ~~ cen 6285    ~<_ cdom 6286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-en 6288  df-dom 6289
This theorem is referenced by:  domeng  6299  php5dom  6398
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