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Theorem inf3lem1 8046
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8053 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 5866 . . 3  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
2 suceq 4943 . . . 4  |-  ( v  =  (/)  ->  suc  v  =  suc  (/) )
32fveq2d 5870 . . 3  |-  ( v  =  (/)  ->  ( F `
 suc  v )  =  ( F `  suc  (/) ) )
41, 3sseq12d 3533 . 2  |-  ( v  =  (/)  ->  ( ( F `  v ) 
C_  ( F `  suc  v )  <->  ( F `  (/) )  C_  ( F `  suc  (/) ) ) )
5 fveq2 5866 . . 3  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
6 suceq 4943 . . . 4  |-  ( v  =  u  ->  suc  v  =  suc  u )
76fveq2d 5870 . . 3  |-  ( v  =  u  ->  ( F `  suc  v )  =  ( F `  suc  u ) )
85, 7sseq12d 3533 . 2  |-  ( v  =  u  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  u )  C_  ( F `  suc  u ) ) )
9 fveq2 5866 . . 3  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
10 suceq 4943 . . . 4  |-  ( v  =  suc  u  ->  suc  v  =  suc  suc  u )
1110fveq2d 5870 . . 3  |-  ( v  =  suc  u  -> 
( F `  suc  v )  =  ( F `  suc  suc  u ) )
129, 11sseq12d 3533 . 2  |-  ( v  =  suc  u  -> 
( ( F `  v )  C_  ( F `  suc  v )  <-> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
13 fveq2 5866 . . 3  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
14 suceq 4943 . . . 4  |-  ( v  =  A  ->  suc  v  =  suc  A )
1514fveq2d 5870 . . 3  |-  ( v  =  A  ->  ( F `  suc  v )  =  ( F `  suc  A ) )
1613, 15sseq12d 3533 . 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 8043 . . 3  |-  ( F `
 (/) )  =  (/)
21 0ss 3814 . . 3  |-  (/)  C_  ( F `  suc  (/) )
2220, 21eqsstri 3534 . 2  |-  ( F `
 (/) )  C_  ( F `  suc  (/) )
23 sstr2 3511 . . . . . . . 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 3116 . . . . . . . . . 10  |-  u  e. 
_V
2717, 18, 26, 19inf3lemc 8044 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  u )  =  ( G `  ( F `  u ) ) )
2827eleq2d 2537 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  v  e.  ( G `  ( F `  u ) ) ) )
29 vex 3116 . . . . . . . . 9  |-  v  e. 
_V
30 fvex 5876 . . . . . . . . 9  |-  ( F `
 u )  e. 
_V
3117, 18, 29, 30inf3lema 8042 . . . . . . . 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 6701 . . . . . . . . . 10  |-  ( u  e.  om  <->  suc  u  e. 
om )
3426sucex 6631 . . . . . . . . . . 11  |-  suc  u  e.  _V
3517, 18, 34, 19inf3lemc 8044 . . . . . . . . . 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 2537 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
v  e.  ( G `
 ( F `  suc  u ) ) ) )
38 fvex 5876 . . . . . . . . 9  |-  ( F `
 suc  u )  e.  _V
3917, 18, 29, 38inf3lema 8042 . . . . . . . 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 3510 . . 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 6711 1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785    |-> cmpt 4505   suc csuc 4880    |` cres 5001   ` cfv 5588   omcom 6685   reccrdg 7076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-recs 7043  df-rdg 7077
This theorem is referenced by:  inf3lem4  8049
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