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Theorem inf3lemd 7344
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 for detailed description. (Contributed by NM, 28-Oct-1996.)
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
inf3lemd  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
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 inf3lemd
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3496 . . . 4  |-  (/)  C_  x
2 fveq2 5541 . . . . . 6  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (/) ) )
3 inf3lem.1 . . . . . . 7  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
4 inf3lem.2 . . . . . . 7  |-  F  =  ( rec ( G ,  (/) )  |`  om )
5 inf3lem.3 . . . . . . 7  |-  A  e. 
_V
6 inf3lem.4 . . . . . . 7  |-  B  e. 
_V
73, 4, 5, 6inf3lemb 7342 . . . . . 6  |-  ( F `
 (/) )  =  (/)
82, 7syl6eq 2344 . . . . 5  |-  ( A  =  (/)  ->  ( F `
 A )  =  (/) )
98sseq1d 3218 . . . 4  |-  ( A  =  (/)  ->  ( ( F `  A ) 
C_  x  <->  (/)  C_  x
) )
101, 9mpbiri 224 . . 3  |-  ( A  =  (/)  ->  ( F `
 A )  C_  x )
1110a1d 22 . 2  |-  ( A  =  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x ) )
12 nnsuc 4689 . . . 4  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. v  e.  om  A  =  suc  v )
13 vex 2804 . . . . . . . . . 10  |-  v  e. 
_V
143, 4, 13, 6inf3lemc 7343 . . . . . . . . 9  |-  ( v  e.  om  ->  ( F `  suc  v )  =  ( G `  ( F `  v ) ) )
1514eleq2d 2363 . . . . . . . 8  |-  ( v  e.  om  ->  (
u  e.  ( F `
 suc  v )  <->  u  e.  ( G `  ( F `  v ) ) ) )
16 vex 2804 . . . . . . . . . 10  |-  u  e. 
_V
17 fvex 5555 . . . . . . . . . 10  |-  ( F `
 v )  e. 
_V
183, 4, 16, 17inf3lema 7341 . . . . . . . . 9  |-  ( u  e.  ( G `  ( F `  v ) )  <->  ( u  e.  x  /\  ( u  i^i  x )  C_  ( F `  v ) ) )
1918simplbi 446 . . . . . . . 8  |-  ( u  e.  ( G `  ( F `  v ) )  ->  u  e.  x )
2015, 19syl6bi 219 . . . . . . 7  |-  ( v  e.  om  ->  (
u  e.  ( F `
 suc  v )  ->  u  e.  x ) )
2120ssrdv 3198 . . . . . 6  |-  ( v  e.  om  ->  ( F `  suc  v ) 
C_  x )
22 fveq2 5541 . . . . . . 7  |-  ( A  =  suc  v  -> 
( F `  A
)  =  ( F `
 suc  v )
)
2322sseq1d 3218 . . . . . 6  |-  ( A  =  suc  v  -> 
( ( F `  A )  C_  x  <->  ( F `  suc  v
)  C_  x )
)
2421, 23syl5ibrcom 213 . . . . 5  |-  ( v  e.  om  ->  ( A  =  suc  v  -> 
( F `  A
)  C_  x )
)
2524rexlimiv 2674 . . . 4  |-  ( E. v  e.  om  A  =  suc  v  ->  ( F `  A )  C_  x )
2612, 25syl 15 . . 3  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  ( F `  A )  C_  x )
2726expcom 424 . 2  |-  ( A  =/=  (/)  ->  ( A  e.  om  ->  ( F `  A )  C_  x
) )
2811, 27pm2.61ine 2535 1  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468    e. cmpt 4093   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  inf3lem2  7346  inf3lem3  7347  inf3lem6  7350
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-pow 4204  ax-pr 4230  ax-un 4528
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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