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Theorem inf3lem1 7583
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7590 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 5728 . . 3  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
2 suceq 4646 . . . 4  |-  ( v  =  (/)  ->  suc  v  =  suc  (/) )
32fveq2d 5732 . . 3  |-  ( v  =  (/)  ->  ( F `
 suc  v )  =  ( F `  suc  (/) ) )
41, 3sseq12d 3377 . 2  |-  ( v  =  (/)  ->  ( ( F `  v ) 
C_  ( F `  suc  v )  <->  ( F `  (/) )  C_  ( F `  suc  (/) ) ) )
5 fveq2 5728 . . 3  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
6 suceq 4646 . . . 4  |-  ( v  =  u  ->  suc  v  =  suc  u )
76fveq2d 5732 . . 3  |-  ( v  =  u  ->  ( F `  suc  v )  =  ( F `  suc  u ) )
85, 7sseq12d 3377 . 2  |-  ( v  =  u  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  u )  C_  ( F `  suc  u ) ) )
9 fveq2 5728 . . 3  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
10 suceq 4646 . . . 4  |-  ( v  =  suc  u  ->  suc  v  =  suc  suc  u )
1110fveq2d 5732 . . 3  |-  ( v  =  suc  u  -> 
( F `  suc  v )  =  ( F `  suc  suc  u ) )
129, 11sseq12d 3377 . 2  |-  ( v  =  suc  u  -> 
( ( F `  v )  C_  ( F `  suc  v )  <-> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
13 fveq2 5728 . . 3  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
14 suceq 4646 . . . 4  |-  ( v  =  A  ->  suc  v  =  suc  A )
1514fveq2d 5732 . . 3  |-  ( v  =  A  ->  ( F `  suc  v )  =  ( F `  suc  A ) )
1613, 15sseq12d 3377 . 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 7580 . . 3  |-  ( F `
 (/) )  =  (/)
21 0ss 3656 . . 3  |-  (/)  C_  ( F `  suc  (/) )
2220, 21eqsstri 3378 . 2  |-  ( F `
 (/) )  C_  ( F `  suc  (/) )
23 sstr2 3355 . . . . . . . 8  |-  ( ( v  i^i  x ) 
C_  ( F `  u )  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  i^i  x
)  C_  ( F `  suc  u ) ) )
2423com12 29 . . . . . . 7  |-  ( ( F `  u ) 
C_  ( F `  suc  u )  ->  (
( v  i^i  x
)  C_  ( F `  u )  ->  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
2524anim2d 549 . . . . . 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 2959 . . . . . . . . . 10  |-  u  e. 
_V
2717, 18, 26, 19inf3lemc 7581 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  u )  =  ( G `  ( F `  u ) ) )
2827eleq2d 2503 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  v  e.  ( G `  ( F `  u ) ) ) )
29 vex 2959 . . . . . . . . 9  |-  v  e. 
_V
30 fvex 5742 . . . . . . . . 9  |-  ( F `
 u )  e. 
_V
3117, 18, 29, 30inf3lema 7579 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  u ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  u ) ) )
3228, 31syl6bb 253 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  ( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  u ) ) ) )
33 peano2b 4861 . . . . . . . . . 10  |-  ( u  e.  om  <->  suc  u  e. 
om )
3426sucex 4791 . . . . . . . . . . 11  |-  suc  u  e.  _V
3517, 18, 34, 19inf3lemc 7581 . . . . . . . . . 10  |-  ( suc  u  e.  om  ->  ( F `  suc  suc  u )  =  ( G `  ( F `
 suc  u )
) )
3633, 35sylbi 188 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  suc  u
)  =  ( G `
 ( F `  suc  u ) ) )
3736eleq2d 2503 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
v  e.  ( G `
 ( F `  suc  u ) ) ) )
38 fvex 5742 . . . . . . . . 9  |-  ( F `
 suc  u )  e.  _V
3917, 18, 29, 38inf3lema 7579 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  suc  u
) )  <->  ( v  e.  x  /\  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
4037, 39syl6bb 253 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  suc  u ) ) ) )
4132, 40imbi12d 312 . . . . . 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 213 . . . . 5  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) ) )
4342imp 419 . . . 4  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) )
4443ssrdv 3354 . . 3  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) )
4544ex 424 . 2  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
464, 8, 12, 16, 22, 45finds 4871 1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628    e. cmpt 4266   suc csuc 4583   omcom 4845    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  inf3lem4  7586
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668
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