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Theorem inf3lem3 7347
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 7352 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 7325. (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 7344 . . . 4  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
61, 2, 3, 4inf3lem2 7346 . . . . 5  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  x ) )
76com12 27 . . . 4  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  x ) )
8 pssdifn0 3528 . . . 4  |-  ( ( ( F `  A
)  C_  x  /\  ( F `  A )  =/=  x )  -> 
( x  \  ( F `  A )
)  =/=  (/) )
95, 7, 8ee12an 1353 . . 3  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( x  \  ( F `  A )
)  =/=  (/) ) )
10 vex 2804 . . . . . 6  |-  x  e. 
_V
11 difss 3316 . . . . . 6  |-  ( x 
\  ( F `  A ) )  C_  x
1210, 11ssexi 4175 . . . . 5  |-  ( x 
\  ( F `  A ) )  e. 
_V
1312zfreg 7325 . . . 4  |-  ( ( x  \  ( F `
 A ) )  =/=  (/)  ->  E. v  e.  ( x  \  ( F `  A )
) ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
14 eldifi 3311 . . . . . . . . . . 11  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  v  e.  x )
15 inssdif0 3534 . . . . . . . . . . . 12  |-  ( ( v  i^i  x ) 
C_  ( F `  A )  <->  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
1615biimpri 197 . . . . . . . . . . 11  |-  ( ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( v  i^i  x )  C_  ( F `  A ) )
1714, 16anim12i 549 . . . . . . . . . 10  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  (
v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  A ) ) )
18 vex 2804 . . . . . . . . . . 11  |-  v  e. 
_V
19 fvex 5555 . . . . . . . . . . 11  |-  ( F `
 A )  e. 
_V
201, 2, 18, 19inf3lema 7341 . . . . . . . . . 10  |-  ( v  e.  ( G `  ( F `  A ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  A ) ) )
2117, 20sylibr 203 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  v  e.  ( G `  ( F `  A )
) )
221, 2, 3, 4inf3lemc 7343 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
2322eleq2d 2363 . . . . . . . . 9  |-  ( A  e.  om  ->  (
v  e.  ( F `
 suc  A )  <->  v  e.  ( G `  ( F `  A ) ) ) )
2421, 23syl5ibr 212 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  v  e.  ( F `  suc  A
) ) )
25 eldifn 3312 . . . . . . . . . 10  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  -.  v  e.  ( F `  A ) )
2625adantr 451 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  -.  v  e.  ( F `  A ) )
2726a1i 10 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  -.  v  e.  ( F `  A ) ) )
2824, 27jcad 519 . . . . . . 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 2357 . . . . . . . . . 10  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 A )  <->  v  e.  ( F `  suc  A
) ) )
3029biimprd 214 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) ) )
31 iman 413 . . . . . . . . 9  |-  ( ( v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) )  <->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3230, 31sylib 188 . . . . . . . 8  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3332necon2ai 2504 . . . . . . 7  |-  ( ( v  e.  ( F `
 suc  A )  /\  -.  v  e.  ( F `  A ) )  ->  ( F `  A )  =/=  ( F `  suc  A ) )
3428, 33syl6 29 . . . . . 6  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
3534exp3a 425 . . . . 5  |-  ( A  e.  om  ->  (
v  e.  ( x 
\  ( F `  A ) )  -> 
( ( v  i^i  ( x  \  ( F `  A )
) )  =  (/)  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) ) )
3635rexlimdv 2679 . . . 4  |-  ( A  e.  om  ->  ( E. v  e.  (
x  \  ( F `  A ) ) ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( F `
 A )  =/=  ( F `  suc  A ) ) )
3713, 36syl5 28 . . 3  |-  ( A  e.  om  ->  (
( x  \  ( F `  A )
)  =/=  (/)  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
389, 37syld 40 . 2  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  ( F `
 suc  A )
) )
3938com12 27 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843    e. cmpt 4093   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  inf3lem4  7348
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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