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Theorem domeng 6809
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
StepHypRef Expression
1 breq2 3967 . 2  |-  ( y  =  B  ->  ( A  ~<_  y  <->  A  ~<_  B ) )
2 sseq2 3142 . . . 4  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
32anbi2d 687 . . 3  |-  ( y  =  B  ->  (
( A  ~~  x  /\  x  C_  y )  <-> 
( A  ~~  x  /\  x  C_  B ) ) )
43exbidv 2006 . 2  |-  ( y  =  B  ->  ( E. x ( A  ~~  x  /\  x  C_  y
)  <->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
5 vex 2743 . . 3  |-  y  e. 
_V
65domen 6808 . 2  |-  ( A  ~<_  y  <->  E. x ( A 
~~  x  /\  x  C_  y ) )
71, 4, 6vtoclbg 2795 1  |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    C_ wss 3094   class class class wbr 3963    ~~ cen 6793    ~<_ cdom 6794
This theorem is referenced by:  undom  6883  mapdom1  6959  mapdom2  6965  domfi  7017  isfinite2  7048  unxpwdom  7236  domfin4  7870  pwfseq  8219  grudomon  8372  ufldom  17584  erdsze2lem1  23071
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-xp 4640  df-rel 4641  df-cnv 4642  df-dm 4644  df-rn 4645  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-en 6797  df-dom 6798
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