MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  domeng Unicode version

Theorem domeng 7081
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 4176 . 2  |-  ( y  =  B  ->  ( A  ~<_  y  <->  A  ~<_  B ) )
2 sseq2 3330 . . . 4  |-  ( y  =  B  ->  (
x  C_  y  <->  x  C_  B
) )
32anbi2d 685 . . 3  |-  ( y  =  B  ->  (
( A  ~~  x  /\  x  C_  y )  <-> 
( A  ~~  x  /\  x  C_  B ) ) )
43exbidv 1633 . 2  |-  ( y  =  B  ->  ( E. x ( A  ~~  x  /\  x  C_  y
)  <->  E. x ( A 
~~  x  /\  x  C_  B ) ) )
5 vex 2919 . . 3  |-  y  e. 
_V
65domen 7080 . 2  |-  ( A  ~<_  y  <->  E. x ( A 
~~  x  /\  x  C_  y ) )
71, 4, 6vtoclbg 2972 1  |-  ( B  e.  C  ->  ( A  ~<_  B  <->  E. x
( A  ~~  x  /\  x  C_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    C_ wss 3280   class class class wbr 4172    ~~ cen 7065    ~<_ cdom 7066
This theorem is referenced by:  undom  7155  mapdom1  7231  mapdom2  7237  domfi  7289  isfinite2  7324  unxpwdom  7513  domfin4  8147  pwfseq  8495  grudomon  8648  ufldom  17947  erdsze2lem1  24842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7069  df-dom 7070
  Copyright terms: Public domain W3C validator