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Theorem domeng 6892
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 4043 . 2  |-  ( y  =  B  ->  ( A  ~<_  y  <->  A  ~<_  B ) )
2 sseq2 3213 . . . 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 1616 . 2  |-  ( y  =  B  ->  ( E. x ( A  ~~  x  /\  x  C_  y
)  <->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
5 vex 2804 . . 3  |-  y  e. 
_V
65domen 6891 . 2  |-  ( A  ~<_  y  <->  E. x ( A 
~~  x  /\  x  C_  y ) )
71, 4, 6vtoclbg 2857 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 1531    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  undom  6966  mapdom1  7042  mapdom2  7048  domfi  7100  isfinite2  7131  unxpwdom  7319  domfin4  7953  pwfseq  8302  grudomon  8455  ufldom  17673  erdsze2lem1  23749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6880  df-dom 6881
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