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Theorem inf3lem5 7349
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 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 4682 . . . 4  |-  ( ( B  e.  A  /\  A  e.  om )  ->  B  e.  om )
21ancoms 439 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  B  e.  om )
3 nnord 4680 . . . . . . 7  |-  ( A  e.  om  ->  Ord  A )
4 ordsucss 4625 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
53, 4syl 15 . . . . . 6  |-  ( A  e.  om  ->  ( B  e.  A  ->  suc 
B  C_  A )
)
65adantr 451 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  suc  B  C_  A
) )
7 peano2b 4688 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 fveq2 5541 . . . . . . . . . 10  |-  ( v  =  suc  B  -> 
( F `  v
)  =  ( F `
 suc  B )
)
98psseq2d 3282 . . . . . . . . 9  |-  ( v  =  suc  B  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  B ) ) )
109imbi2d 307 . . . . . . . 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 5541 . . . . . . . . . 10  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
1211psseq2d 3282 . . . . . . . . 9  |-  ( v  =  u  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  u )
) )
1312imbi2d 307 . . . . . . . 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 5541 . . . . . . . . . 10  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
1514psseq2d 3282 . . . . . . . . 9  |-  ( v  =  suc  u  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  u ) ) )
1615imbi2d 307 . . . . . . . 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 5541 . . . . . . . . . 10  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
1817psseq2d 3282 . . . . . . . . 9  |-  ( v  =  A  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  A )
) )
1918imbi2d 307 . . . . . . . 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 7348 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( B  e.  om  ->  ( F `  B
)  C.  ( F `  suc  B ) ) )
2423com12 27 . . . . . . . . 9  |-  ( B  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
257, 24sylbir 204 . . . . . . . 8  |-  ( suc 
B  e.  om  ->  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
26 vex 2804 . . . . . . . . . . . 12  |-  u  e. 
_V
2720, 21, 26, 22inf3lem4 7348 . . . . . . . . . . 11  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( u  e.  om  ->  ( F `  u
)  C.  ( F `  suc  u ) ) )
28 psstr 3293 . . . . . . . . . . . 12  |-  ( ( ( F `  B
)  C.  ( F `  u )  /\  ( F `  u )  C.  ( F `  suc  u ) )  -> 
( F `  B
)  C.  ( F `  suc  u ) )
2928expcom 424 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  suc  u )  ->  (
( F `  B
)  C.  ( F `  u )  ->  ( F `  B )  C.  ( F `  suc  u ) ) )
3027, 29syl6com 31 . . . . . . . . . 10  |-  ( u  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( F `  B )  C.  ( F `  u )  ->  ( F `  B
)  C.  ( F `  suc  u ) ) ) )
3130a2d 23 . . . . . . . . 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 706 . . . . . . . 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 4699 . . . . . . 7  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  suc  B  C_  A )  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  A ) ) )
3433ex 423 . . . . . 6  |-  ( ( A  e.  om  /\  suc  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
357, 34sylan2b 461 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
366, 35syld 40 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) ) )
3736impancom 427 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( B  e.  om  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) ) )
382, 37mpd 14 . 2  |-  ( ( A  e.  om  /\  B  e.  A )  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) )
3938com12 27 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165    C. wpss 3166   (/)c0 3468   U.cuni 3843    e. cmpt 4093   Ord word 4407   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  inf3lem6  7350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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