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Theorem inf3lem5 8146
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8149 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 6716 . . . 4  |-  ( ( B  e.  A  /\  A  e.  om )  ->  B  e.  om )
21ancoms 454 . . 3  |-  ( ( A  e.  om  /\  B  e.  A )  ->  B  e.  om )
3 nnord 6714 . . . . . . 7  |-  ( A  e.  om  ->  Ord  A )
4 ordsucss 6659 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
53, 4syl 17 . . . . . 6  |-  ( A  e.  om  ->  ( B  e.  A  ->  suc 
B  C_  A )
)
65adantr 466 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  suc  B  C_  A
) )
7 peano2b 6722 . . . . . 6  |-  ( B  e.  om  <->  suc  B  e. 
om )
8 fveq2 5881 . . . . . . . . . 10  |-  ( v  =  suc  B  -> 
( F `  v
)  =  ( F `
 suc  B )
)
98psseq2d 3558 . . . . . . . . 9  |-  ( v  =  suc  B  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  B ) ) )
109imbi2d 317 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( v  =  u  ->  ( F `  v )  =  ( F `  u ) )
1211psseq2d 3558 . . . . . . . . 9  |-  ( v  =  u  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  u )
) )
1312imbi2d 317 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( v  =  suc  u  -> 
( F `  v
)  =  ( F `
 suc  u )
)
1514psseq2d 3558 . . . . . . . . 9  |-  ( v  =  suc  u  -> 
( ( F `  B )  C.  ( F `  v )  <->  ( F `  B ) 
C.  ( F `  suc  u ) ) )
1615imbi2d 317 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( v  =  A  ->  ( F `  v )  =  ( F `  A ) )
1817psseq2d 3558 . . . . . . . . 9  |-  ( v  =  A  ->  (
( F `  B
)  C.  ( F `  v )  <->  ( F `  B )  C.  ( F `  A )
) )
1918imbi2d 317 . . . . . . . 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 8145 . . . . . . . . . 10  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( B  e.  om  ->  ( F `  B
)  C.  ( F `  suc  B ) ) )
2423com12 32 . . . . . . . . 9  |-  ( B  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
257, 24sylbir 216 . . . . . . . 8  |-  ( suc 
B  e.  om  ->  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  suc  B ) ) )
26 vex 3083 . . . . . . . . . . . 12  |-  u  e. 
_V
2720, 21, 26, 22inf3lem4 8145 . . . . . . . . . . 11  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( u  e.  om  ->  ( F `  u
)  C.  ( F `  suc  u ) ) )
28 psstr 3569 . . . . . . . . . . . 12  |-  ( ( ( F `  B
)  C.  ( F `  u )  /\  ( F `  u )  C.  ( F `  suc  u ) )  -> 
( F `  B
)  C.  ( F `  suc  u ) )
2928expcom 436 . . . . . . . . . . 11  |-  ( ( F `  u ) 
C.  ( F `  suc  u )  ->  (
( F `  B
)  C.  ( F `  u )  ->  ( F `  B )  C.  ( F `  suc  u ) ) )
3027, 29syl6com 36 . . . . . . . . . 10  |-  ( u  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( F `  B )  C.  ( F `  u )  ->  ( F `  B
)  C.  ( F `  suc  u ) ) ) )
3130a2d 29 . . . . . . . . 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 730 . . . . . . . 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 6734 . . . . . . 7  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  suc  B  C_  A )  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  B
)  C.  ( F `  A ) ) )
3433ex 435 . . . . . 6  |-  ( ( A  e.  om  /\  suc  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
357, 34sylan2b 477 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  B  C_  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x )  ->  ( F `  B )  C.  ( F `  A
) ) ) )
366, 35syld 45 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  A  ->  ( ( x  =/=  (/)  /\  x  C_  U. x
)  ->  ( F `  B )  C.  ( F `  A )
) ) )
3736impancom 441 . . 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 32 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( ( A  e. 
om  /\  B  e.  A )  ->  ( F `  B )  C.  ( F `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   {crab 2775   _Vcvv 3080    i^i cin 3435    C_ wss 3436    C. wpss 3437   (/)c0 3761   U.cuni 4219    |-> cmpt 4482    |` cres 4855   Ord word 5441   suc csuc 5444   ` cfv 5601   omcom 6706   reccrdg 7138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-reg 8116
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139
This theorem is referenced by:  inf3lem6  8147
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