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Theorem inf3lem3 7585
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7590 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7563. (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
inf3lem3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  ( 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 inf3lem3
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 inf3lem.1 . . . . 5  |-  G  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
2 inf3lem.2 . . . . 5  |-  F  =  ( rec ( G ,  (/) )  |`  om )
3 inf3lem.3 . . . . 5  |-  A  e. 
_V
4 inf3lem.4 . . . . 5  |-  B  e. 
_V
51, 2, 3, 4inf3lemd 7582 . . . 4  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
61, 2, 3, 4inf3lem2 7584 . . . . 5  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  x ) )
76com12 29 . . . 4  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  x ) )
8 pssdifn0 3689 . . . 4  |-  ( ( ( F `  A
)  C_  x  /\  ( F `  A )  =/=  x )  -> 
( x  \  ( F `  A )
)  =/=  (/) )
95, 7, 8ee12an 1372 . . 3  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( x  \  ( F `  A )
)  =/=  (/) ) )
10 vex 2959 . . . . . 6  |-  x  e. 
_V
11 difss 3474 . . . . . 6  |-  ( x 
\  ( F `  A ) )  C_  x
1210, 11ssexi 4348 . . . . 5  |-  ( x 
\  ( F `  A ) )  e. 
_V
1312zfreg 7563 . . . 4  |-  ( ( x  \  ( F `
 A ) )  =/=  (/)  ->  E. v  e.  ( x  \  ( F `  A )
) ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
14 eldifi 3469 . . . . . . . . . . 11  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  v  e.  x )
15 inssdif0 3695 . . . . . . . . . . . 12  |-  ( ( v  i^i  x ) 
C_  ( F `  A )  <->  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
1615biimpri 198 . . . . . . . . . . 11  |-  ( ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( v  i^i  x )  C_  ( F `  A ) )
1714, 16anim12i 550 . . . . . . . . . 10  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  (
v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  A ) ) )
18 vex 2959 . . . . . . . . . . 11  |-  v  e. 
_V
19 fvex 5742 . . . . . . . . . . 11  |-  ( F `
 A )  e. 
_V
201, 2, 18, 19inf3lema 7579 . . . . . . . . . 10  |-  ( v  e.  ( G `  ( F `  A ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  A ) ) )
2117, 20sylibr 204 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  v  e.  ( G `  ( F `  A )
) )
221, 2, 3, 4inf3lemc 7581 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
2322eleq2d 2503 . . . . . . . . 9  |-  ( A  e.  om  ->  (
v  e.  ( F `
 suc  A )  <->  v  e.  ( G `  ( F `  A ) ) ) )
2421, 23syl5ibr 213 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  v  e.  ( F `  suc  A
) ) )
25 eldifn 3470 . . . . . . . . . 10  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  -.  v  e.  ( F `  A ) )
2625adantr 452 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  -.  v  e.  ( F `  A ) )
2726a1i 11 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  -.  v  e.  ( F `  A ) ) )
2824, 27jcad 520 . . . . . . 7  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  ( v  e.  ( F `  suc  A )  /\  -.  v  e.  ( F `  A
) ) ) )
29 eleq2 2497 . . . . . . . . . 10  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 A )  <->  v  e.  ( F `  suc  A
) ) )
3029biimprd 215 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) ) )
31 iman 414 . . . . . . . . 9  |-  ( ( v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) )  <->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3230, 31sylib 189 . . . . . . . 8  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3332necon2ai 2649 . . . . . . 7  |-  ( ( v  e.  ( F `
 suc  A )  /\  -.  v  e.  ( F `  A ) )  ->  ( F `  A )  =/=  ( F `  suc  A ) )
3428, 33syl6 31 . . . . . 6  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
3534exp3a 426 . . . . 5  |-  ( A  e.  om  ->  (
v  e.  ( x 
\  ( F `  A ) )  -> 
( ( v  i^i  ( x  \  ( F `  A )
) )  =  (/)  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) ) )
3635rexlimdv 2829 . . . 4  |-  ( A  e.  om  ->  ( E. v  e.  (
x  \  ( F `  A ) ) ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( F `
 A )  =/=  ( F `  suc  A ) ) )
3713, 36syl5 30 . . 3  |-  ( A  e.  om  ->  (
( x  \  ( F `  A )
)  =/=  (/)  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
389, 37syld 42 . 2  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  ( F `
 suc  A )
) )
3938com12 29 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   U.cuni 4015    e. cmpt 4266   suc csuc 4583   omcom 4845    |` cres 4880   ` cfv 5454   reccrdg 6667
This theorem is referenced by:  inf3lem4  7586
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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