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Theorem domeng 7019
Description: Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
domeng  |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem domeng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 breq2 4129 . 2  |-  ( y  =  B  ->  ( A  ~<_  y  <->  A  ~<_  B ) )
2 sseq2 3286 . . . 4  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
32anbi2d 684 . . 3  |-  ( y  =  B  ->  (
( A  ~~  x  /\  x  C_  y )  <-> 
( A  ~~  x  /\  x  C_  B ) ) )
43exbidv 1631 . 2  |-  ( y  =  B  ->  ( E. x ( A  ~~  x  /\  x  C_  y
)  <->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
5 vex 2876 . . 3  |-  y  e. 
_V
65domen 7018 . 2  |-  ( A  ~<_  y  <->  E. x ( A 
~~  x  /\  x  C_  y ) )
71, 4, 6vtoclbg 2929 1  |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715    C_ wss 3238   class class class wbr 4125    ~~ cen 7003    ~<_ cdom 7004
This theorem is referenced by:  undom  7093  mapdom1  7169  mapdom2  7175  domfi  7227  isfinite2  7262  unxpwdom  7450  domfin4  8084  pwfseq  8433  grudomon  8586  ufldom  17870  erdsze2lem1  24337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-en 7007  df-dom 7008
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