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Theorem inf3lem5 7587
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7590 for detailed description. (Contributed by NM, 29-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
inf3lem5  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( A  e. 
om  /\  B  e.  A )  ->  ( F `  B )  C.  ( F `  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 inf3lem5
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4855 . . . 4  |-  ( ( B  e.  A  /\  A  e.  om )  ->  B  e.  om )
21ancoms 440 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  B  e.  om )
3 nnord 4853 . . . . . . 7  |-  ( A  e.  om  ->  Ord  A )
4 ordsucss 4798 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
53, 4syl 16 . . . . . 6  |-  ( A  e.  om  ->  ( B  e.  A  ->  suc 
B  C_  A )
)
65adantr 452 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  suc  B  C_  A
) )
7 peano2b 4861 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 fveq2 5728 . . . . . . . . . 10  |-  ( v  =  suc  B  -> 
( F `  v
)  =  ( F `
 suc  B )
)
98psseq2d 3440 . . . . . . . . 9  |-  ( v  =  suc  B  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  B ) ) )
109imbi2d 308 . . . . . . . 8  |-  ( v  =  suc  B  -> 
( ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  v
) )  <->  ( (
x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) ) )
11 fveq2 5728 . . . . . . . . . 10  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
1211psseq2d 3440 . . . . . . . . 9  |-  ( v  =  u  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  u )
) )
1312imbi2d 308 . . . . . . . 8  |-  ( v  =  u  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  v )
)  <->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  u
) ) ) )
14 fveq2 5728 . . . . . . . . . 10  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
1514psseq2d 3440 . . . . . . . . 9  |-  ( v  =  suc  u  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  u ) ) )
1615imbi2d 308 . . . . . . . 8  |-  ( v  =  suc  u  -> 
( ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  v
) )  <->  ( (
x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  u ) ) ) )
17 fveq2 5728 . . . . . . . . . 10  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
1817psseq2d 3440 . . . . . . . . 9  |-  ( v  =  A  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  A )
) )
1918imbi2d 308 . . . . . . . 8  |-  ( v  =  A  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  v )
)  <->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
20 inf3lem.1 . . . . . . . . . . 11  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
21 inf3lem.2 . . . . . . . . . . 11  |-  F  =  ( rec ( G ,  (/) )  |`  om )
22 inf3lem.4 . . . . . . . . . . 11  |-  B  e. 
_V
2320, 21, 22, 22inf3lem4 7586 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( B  e.  om  ->  ( F `  B
)  C.  ( F `  suc  B ) ) )
2423com12 29 . . . . . . . . 9  |-  ( B  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
257, 24sylbir 205 . . . . . . . 8  |-  ( suc 
B  e.  om  ->  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
26 vex 2959 . . . . . . . . . . . 12  |-  u  e. 
_V
2720, 21, 26, 22inf3lem4 7586 . . . . . . . . . . 11  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( u  e.  om  ->  ( F `  u
)  C.  ( F `  suc  u ) ) )
28 psstr 3451 . . . . . . . . . . . 12  |-  ( ( ( F `  B
)  C.  ( F `  u )  /\  ( F `  u )  C.  ( F `  suc  u ) )  -> 
( F `  B
)  C.  ( F `  suc  u ) )
2928expcom 425 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  suc  u )  ->  (
( F `  B
)  C.  ( F `  u )  ->  ( F `  B )  C.  ( F `  suc  u ) ) )
3027, 29syl6com 33 . . . . . . . . . 10  |-  ( u  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( F `  B )  C.  ( F `  u )  ->  ( F `  B
)  C.  ( F `  suc  u ) ) ) )
3130a2d 24 . . . . . . . . 9  |-  ( u  e.  om  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  u )
)  ->  ( (
x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  u ) ) ) )
3231ad2antrr 707 . . . . . . . 8  |-  ( ( ( u  e.  om  /\ 
suc  B  e.  om )  /\  suc  B  C_  u )  ->  (
( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  u )
)  ->  ( (
x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  u ) ) ) )
3310, 13, 16, 19, 25, 32findsg 4872 . . . . . . 7  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  suc  B  C_  A )  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  A ) ) )
3433ex 424 . . . . . 6  |-  ( ( A  e.  om  /\  suc  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
357, 34sylan2b 462 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
366, 35syld 42 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) ) )
3736impancom 428 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( B  e.  om  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) ) )
382, 37mpd 15 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) )
3938com12 29 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( A  e. 
om  /\  B  e.  A )  ->  ( F `  B )  C.  ( F `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320    C. wpss 3321   (/)c0 3628   U.cuni 4015    e. cmpt 4266   Ord word 4580   suc csuc 4583   omcom 4845    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  inf3lem6  7588
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  ax-reg 7560
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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