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Theorem inf3lem1 8045
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8052 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 5853 . . 3  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
2 suceq 4930 . . . 4  |-  ( v  =  (/)  ->  suc  v  =  suc  (/) )
32fveq2d 5857 . . 3  |-  ( v  =  (/)  ->  ( F `
 suc  v )  =  ( F `  suc  (/) ) )
41, 3sseq12d 3516 . 2  |-  ( v  =  (/)  ->  ( ( F `  v ) 
C_  ( F `  suc  v )  <->  ( F `  (/) )  C_  ( F `  suc  (/) ) ) )
5 fveq2 5853 . . 3  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
6 suceq 4930 . . . 4  |-  ( v  =  u  ->  suc  v  =  suc  u )
76fveq2d 5857 . . 3  |-  ( v  =  u  ->  ( F `  suc  v )  =  ( F `  suc  u ) )
85, 7sseq12d 3516 . 2  |-  ( v  =  u  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  u )  C_  ( F `  suc  u ) ) )
9 fveq2 5853 . . 3  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
10 suceq 4930 . . . 4  |-  ( v  =  suc  u  ->  suc  v  =  suc  suc  u )
1110fveq2d 5857 . . 3  |-  ( v  =  suc  u  -> 
( F `  suc  v )  =  ( F `  suc  suc  u ) )
129, 11sseq12d 3516 . 2  |-  ( v  =  suc  u  -> 
( ( F `  v )  C_  ( F `  suc  v )  <-> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
13 fveq2 5853 . . 3  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
14 suceq 4930 . . . 4  |-  ( v  =  A  ->  suc  v  =  suc  A )
1514fveq2d 5857 . . 3  |-  ( v  =  A  ->  ( F `  suc  v )  =  ( F `  suc  A ) )
1613, 15sseq12d 3516 . 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 8042 . . 3  |-  ( F `
 (/) )  =  (/)
21 0ss 3797 . . 3  |-  (/)  C_  ( F `  suc  (/) )
2220, 21eqsstri 3517 . 2  |-  ( F `
 (/) )  C_  ( F `  suc  (/) )
23 sstr2 3494 . . . . . . . 8  |-  ( ( v  i^i  x ) 
C_  ( F `  u )  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  i^i  x
)  C_  ( F `  suc  u ) ) )
2423com12 31 . . . . . . 7  |-  ( ( F `  u ) 
C_  ( F `  suc  u )  ->  (
( v  i^i  x
)  C_  ( F `  u )  ->  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
2524anim2d 565 . . . . . 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 3096 . . . . . . . . . 10  |-  u  e. 
_V
2717, 18, 26, 19inf3lemc 8043 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  u )  =  ( G `  ( F `  u ) ) )
2827eleq2d 2511 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  v  e.  ( G `  ( F `  u ) ) ) )
29 vex 3096 . . . . . . . . 9  |-  v  e. 
_V
30 fvex 5863 . . . . . . . . 9  |-  ( F `
 u )  e. 
_V
3117, 18, 29, 30inf3lema 8041 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  u ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  u ) ) )
3228, 31syl6bb 261 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  ( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  u ) ) ) )
33 peano2b 6698 . . . . . . . . . 10  |-  ( u  e.  om  <->  suc  u  e. 
om )
3426sucex 6628 . . . . . . . . . . 11  |-  suc  u  e.  _V
3517, 18, 34, 19inf3lemc 8043 . . . . . . . . . 10  |-  ( suc  u  e.  om  ->  ( F `  suc  suc  u )  =  ( G `  ( F `
 suc  u )
) )
3633, 35sylbi 195 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  suc  u
)  =  ( G `
 ( F `  suc  u ) ) )
3736eleq2d 2511 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
v  e.  ( G `
 ( F `  suc  u ) ) ) )
38 fvex 5863 . . . . . . . . 9  |-  ( F `
 suc  u )  e.  _V
3917, 18, 29, 38inf3lema 8041 . . . . . . . 8  |-  ( v  e.  ( G `  ( F `  suc  u
) )  <->  ( v  e.  x  /\  (
v  i^i  x )  C_  ( F `  suc  u ) ) )
4037, 39syl6bb 261 . . . . . . 7  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
( v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  suc  u ) ) ) )
4132, 40imbi12d 320 . . . . . 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 221 . . . . 5  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) ) )
4342imp 429 . . . 4  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) )
4443ssrdv 3493 . . 3  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) )
4544ex 434 . 2  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
464, 8, 12, 16, 22, 45finds 6708 1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   {crab 2795   _Vcvv 3093    i^i cin 3458    C_ wss 3459   (/)c0 3768    |-> cmpt 4492   suc csuc 4867    |` cres 4988   ` cfv 5575   omcom 6682   reccrdg 7074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-om 6683  df-recs 7041  df-rdg 7075
This theorem is referenced by:  inf3lem4  8048
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