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Theorem inf3lem1 7977
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7984 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 5787 . . 3  |-  ( v  =  (/)  ->  ( F `
 v )  =  ( F `  (/) ) )
2 suceq 4870 . . . 4  |-  ( v  =  (/)  ->  suc  v  =  suc  (/) )
32fveq2d 5791 . . 3  |-  ( v  =  (/)  ->  ( F `
 suc  v )  =  ( F `  suc  (/) ) )
41, 3sseq12d 3459 . 2  |-  ( v  =  (/)  ->  ( ( F `  v ) 
C_  ( F `  suc  v )  <->  ( F `  (/) )  C_  ( F `  suc  (/) ) ) )
5 fveq2 5787 . . 3  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
6 suceq 4870 . . . 4  |-  ( v  =  u  ->  suc  v  =  suc  u )
76fveq2d 5791 . . 3  |-  ( v  =  u  ->  ( F `  suc  v )  =  ( F `  suc  u ) )
85, 7sseq12d 3459 . 2  |-  ( v  =  u  ->  (
( F `  v
)  C_  ( F `  suc  v )  <->  ( F `  u )  C_  ( F `  suc  u ) ) )
9 fveq2 5787 . . 3  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
10 suceq 4870 . . . 4  |-  ( v  =  suc  u  ->  suc  v  =  suc  suc  u )
1110fveq2d 5791 . . 3  |-  ( v  =  suc  u  -> 
( F `  suc  v )  =  ( F `  suc  suc  u ) )
129, 11sseq12d 3459 . 2  |-  ( v  =  suc  u  -> 
( ( F `  v )  C_  ( F `  suc  v )  <-> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
13 fveq2 5787 . . 3  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
14 suceq 4870 . . . 4  |-  ( v  =  A  ->  suc  v  =  suc  A )
1514fveq2d 5791 . . 3  |-  ( v  =  A  ->  ( F `  suc  v )  =  ( F `  suc  A ) )
1613, 15sseq12d 3459 . 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 7974 . . 3  |-  ( F `
 (/) )  =  (/)
21 0ss 3754 . . 3  |-  (/)  C_  ( F `  suc  (/) )
2220, 21eqsstri 3460 . 2  |-  ( F `
 (/) )  C_  ( F `  suc  (/) )
23 sstr2 3437 . . . . . . . 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 563 . . . . . 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 3050 . . . . . . . . . 10  |-  u  e. 
_V
2717, 18, 26, 19inf3lemc 7975 . . . . . . . . 9  |-  ( u  e.  om  ->  ( F `  suc  u )  =  ( G `  ( F `  u ) ) )
2827eleq2d 2462 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  u )  <->  v  e.  ( G `  ( F `  u ) ) ) )
29 vex 3050 . . . . . . . . 9  |-  v  e. 
_V
30 fvex 5797 . . . . . . . . 9  |-  ( F `
 u )  e. 
_V
3117, 18, 29, 30inf3lema 7973 . . . . . . . 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 6633 . . . . . . . . . 10  |-  ( u  e.  om  <->  suc  u  e. 
om )
3426sucex 6563 . . . . . . . . . . 11  |-  suc  u  e.  _V
3517, 18, 34, 19inf3lemc 7975 . . . . . . . . . 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 2462 . . . . . . . 8  |-  ( u  e.  om  ->  (
v  e.  ( F `
 suc  suc  u )  <-> 
v  e.  ( G `
 ( F `  suc  u ) ) ) )
38 fvex 5797 . . . . . . . . 9  |-  ( F `
 suc  u )  e.  _V
3917, 18, 29, 38inf3lema 7973 . . . . . . . 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 318 . . . . . 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 427 . . . 4  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( v  e.  ( F `  suc  u
)  ->  v  e.  ( F `  suc  suc  u ) ) )
4443ssrdv 3436 . . 3  |-  ( ( u  e.  om  /\  ( F `  u ) 
C_  ( F `  suc  u ) )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) )
4544ex 432 . 2  |-  ( u  e.  om  ->  (
( F `  u
)  C_  ( F `  suc  u )  -> 
( F `  suc  u )  C_  ( F `  suc  suc  u
) ) )
464, 8, 12, 16, 22, 45finds 6643 1  |-  ( A  e.  om  ->  ( F `  A )  C_  ( F `  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   {crab 2746   _Vcvv 3047    i^i cin 3401    C_ wss 3402   (/)c0 3724    |-> cmpt 4438   suc csuc 4807    |` cres 4928   ` cfv 5509   omcom 6617   reccrdg 7011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-om 6618  df-recs 6978  df-rdg 7012
This theorem is referenced by:  inf3lem4  7980
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