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Theorem inf3lem7 7351
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 5767. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
Hypotheses
Ref Expression
inf3lem.1  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
inf3lem.2  |-  F  =  ( rec ( G ,  (/) )  |`  om )
inf3lem.3  |-  A  e. 
_V
inf3lem.4  |-  B  e. 
_V
Assertion
Ref Expression
inf3lem7  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
Distinct variable group:    x, y, w
Allowed substitution hints:    A( x, y, w)    B( x, y, w)    F( x, y, w)    G( x, y, w)

Proof of Theorem inf3lem7
StepHypRef Expression
1 inf3lem.1 . . 3  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
2 inf3lem.2 . . 3  |-  F  =  ( rec ( G ,  (/) )  |`  om )
3 inf3lem.3 . . 3  |-  A  e. 
_V
4 inf3lem.4 . . 3  |-  B  e. 
_V
51, 2, 3, 4inf3lem6 7350 . 2  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  F : om -1-1-> ~P x
)
6 vex 2804 . . 3  |-  x  e. 
_V
76pwex 4209 . 2  |-  ~P x  e.  _V
8 f1dmex 5767 . 2  |-  ( ( F : om -1-1-> ~P x  /\  ~P x  e. 
_V )  ->  om  e.  _V )
95, 7, 8sylancl 643 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843    e. cmpt 4093   omcom 4672    |` cres 4707   -1-1->wf1 5268   reccrdg 6438
This theorem is referenced by:  inf3  7352  infeq5  7354
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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