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Theorem inf3lem1 7329
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7336 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
inf3lem1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
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 inf3lem1
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
2 suceq 4457 . . . 4  |-  ( v  =  (/)  ->  suc  v  =  suc  (/) )
32fveq2d 5529 . . 3  |-  ( v  =  (/)  ->  ( F `
 suc  v )  =  ( F `  suc  (/) ) )
41, 3sseq12d 3207 . 2  |-  ( v  =  (/)  ->  ( ( F `  v ) 
C_  ( F `  suc  v )  <->  ( F `  (/) )  C_  ( F `  suc  (/) ) ) )
5 fveq2 5525 . . 3  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
6 suceq 4457 . . . 4  |-  ( v  =  u  ->  suc  v  =  suc  u )
76fveq2d 5529 . . 3  |-  ( v  =  u  ->  ( F `  suc  v )  =  ( F `  suc  u ) )
85, 7sseq12d 3207 . 2  |-  ( v  =  u  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  u )  C_  ( F `  suc  u ) ) )
9 fveq2 5525 . . 3  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
10 suceq 4457 . . . 4  |-  ( v  =  suc  u  ->  suc  v  =  suc  suc  u )
1110fveq2d 5529 . . 3  |-  ( v  =  suc  u  -> 
( F `  suc  v )  =  ( F `  suc  suc  u ) )
129, 11sseq12d 3207 . 2  |-  ( v  =  suc  u  -> 
( ( F `  v )  C_  ( F `  suc  v )  <-> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
13 fveq2 5525 . . 3  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
14 suceq 4457 . . . 4  |-  ( v  =  A  ->  suc  v  =  suc  A )
1514fveq2d 5529 . . 3  |-  ( v  =  A  ->  ( F `  suc  v )  =  ( F `  suc  A ) )
1613, 15sseq12d 3207 . 2  |-  ( v  =  A  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  A )  C_  ( F `  suc  A ) ) )
17 inf3lem.1 . . . 4  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
18 inf3lem.2 . . . 4  |-  F  =  ( rec ( G ,  (/) )  |`  om )
19 inf3lem.3 . . . 4  |-  A  e. 
_V
2017, 18, 19, 19inf3lemb 7326 . . 3  |-  ( F `
 (/) )  =  (/)
21 0ss 3483 . . 3  |-  (/)  C_  ( F `  suc  (/) )
2220, 21eqsstri 3208 . 2  |-  ( F `
 (/) )  C_  ( F `  suc  (/) )
23 sstr2 3186 . . . . . . . 8  |-  ( ( v  i^i  x ) 
C_  ( F `  u )  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  i^i  x
)  C_  ( F `  suc  u ) ) )
2423com12 27 . . . . . . 7  |-  ( ( F `  u ) 
C_  ( F `  suc  u )  ->  (
( v  i^i  x
)  C_  ( F `  u )  ->  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
2524anim2d 548 . . . . . 6  |-  ( ( F `  u ) 
C_  ( F `  suc  u )  ->  (
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  u ) )  -> 
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  suc  u ) ) ) )
26 vex 2791 . . . . . . . . . 10  |-  u  e. 
_V
2717, 18, 26, 19inf3lemc 7327 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  u )  =  ( G `  ( F `  u ) ) )
2827eleq2d 2350 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  v  e.  ( G `  ( F `  u ) ) ) )
29 vex 2791 . . . . . . . . 9  |-  v  e. 
_V
30 fvex 5539 . . . . . . . . 9  |-  ( F `
 u )  e. 
_V
3117, 18, 29, 30inf3lema 7325 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  u ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  u ) ) )
3228, 31syl6bb 252 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  ( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  u ) ) ) )
33 peano2b 4672 . . . . . . . . . 10  |-  ( u  e.  om  <->  suc  u  e. 
om )
3426sucex 4602 . . . . . . . . . . 11  |-  suc  u  e.  _V
3517, 18, 34, 19inf3lemc 7327 . . . . . . . . . 10  |-  ( suc  u  e.  om  ->  ( F `  suc  suc  u )  =  ( G `  ( F `
 suc  u )
) )
3633, 35sylbi 187 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  suc  u
)  =  ( G `
 ( F `  suc  u ) ) )
3736eleq2d 2350 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
v  e.  ( G `
 ( F `  suc  u ) ) ) )
38 fvex 5539 . . . . . . . . 9  |-  ( F `
 suc  u )  e.  _V
3917, 18, 29, 38inf3lema 7325 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  suc  u
) )  <->  ( v  e.  x  /\  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
4037, 39syl6bb 252 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  suc  u ) ) ) )
4132, 40imbi12d 311 . . . . . 6  |-  ( u  e.  om  ->  (
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) )  <->  ( (
v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  u ) )  -> 
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  suc  u ) ) ) ) )
4225, 41syl5ibr 212 . . . . 5  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) ) )
4342imp 418 . . . 4  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) )
4443ssrdv 3185 . . 3  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) )
4544ex 423 . 2  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
464, 8, 12, 16, 22, 45finds 4682 1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455    e. cmpt 4077   suc csuc 4394   omcom 4656    |` cres 4691   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  inf3lem4  7332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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