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Theorem domeng 7424
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 4394 . 2  |-  ( y  =  B  ->  ( A  ~<_  y  <->  A  ~<_  B ) )
2 sseq2 3476 . . . 4  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
32anbi2d 703 . . 3  |-  ( y  =  B  ->  (
( A  ~~  x  /\  x  C_  y )  <-> 
( A  ~~  x  /\  x  C_  B ) ) )
43exbidv 1681 . 2  |-  ( y  =  B  ->  ( E. x ( A  ~~  x  /\  x  C_  y
)  <->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
5 vex 3071 . . 3  |-  y  e. 
_V
65domen 7423 . 2  |-  ( A  ~<_  y  <->  E. x ( A 
~~  x  /\  x  C_  y ) )
71, 4, 6vtoclbg 3127 1  |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    C_ wss 3426   class class class wbr 4390    ~~ cen 7407    ~<_ cdom 7408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-xp 4944  df-rel 4945  df-cnv 4946  df-dm 4948  df-rn 4949  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-en 7411  df-dom 7412
This theorem is referenced by:  undom  7499  mapdom1  7576  mapdom2  7582  domfi  7635  isfinite2  7671  unxpwdom  7905  domfin4  8581  pwfseq  8932  grudomon  9085  ufldom  19651  erdsze2lem1  27225
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