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Theorem inf3lem3 8144
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8149 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8119. (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 8141 . . . 4  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
61, 2, 3, 4inf3lem2 8143 . . . . 5  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  x ) )
76com12 32 . . . 4  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  x ) )
8 pssdifn0 3857 . . . 4  |-  ( ( ( F `  A
)  C_  x  /\  ( F `  A )  =/=  x )  -> 
( x  \  ( F `  A )
)  =/=  (/) )
95, 7, 8syl6an 547 . . 3  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( x  \  ( F `  A )
)  =/=  (/) ) )
10 vex 3083 . . . . . 6  |-  x  e. 
_V
11 difss 3592 . . . . . 6  |-  ( x 
\  ( F `  A ) )  C_  x
1210, 11ssexi 4569 . . . . 5  |-  ( x 
\  ( F `  A ) )  e. 
_V
1312zfreg 8119 . . . 4  |-  ( ( x  \  ( F `
 A ) )  =/=  (/)  ->  E. v  e.  ( x  \  ( F `  A )
) ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
14 eldifi 3587 . . . . . . . . . . 11  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  v  e.  x )
15 inssdif0 3864 . . . . . . . . . . . 12  |-  ( ( v  i^i  x ) 
C_  ( F `  A )  <->  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
1615biimpri 209 . . . . . . . . . . 11  |-  ( ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( v  i^i  x )  C_  ( F `  A ) )
1714, 16anim12i 568 . . . . . . . . . 10  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  (
v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  A ) ) )
18 vex 3083 . . . . . . . . . . 11  |-  v  e. 
_V
19 fvex 5891 . . . . . . . . . . 11  |-  ( F `
 A )  e. 
_V
201, 2, 18, 19inf3lema 8138 . . . . . . . . . 10  |-  ( v  e.  ( G `  ( F `  A ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  A ) ) )
2117, 20sylibr 215 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  v  e.  ( G `  ( F `  A )
) )
221, 2, 3, 4inf3lemc 8140 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
2322eleq2d 2492 . . . . . . . . 9  |-  ( A  e.  om  ->  (
v  e.  ( F `
 suc  A )  <->  v  e.  ( G `  ( F `  A ) ) ) )
2421, 23syl5ibr 224 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  v  e.  ( F `  suc  A
) ) )
25 eldifn 3588 . . . . . . . . . 10  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  -.  v  e.  ( F `  A ) )
2625adantr 466 . . . . . . . . 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 535 . . . . . . 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 2496 . . . . . . . . . 10  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 A )  <->  v  e.  ( F `  suc  A
) ) )
3029biimprd 226 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) ) )
31 iman 425 . . . . . . . . 9  |-  ( ( v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) )  <->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3230, 31sylib 199 . . . . . . . 8  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3332necon2ai 2655 . . . . . . 7  |-  ( ( v  e.  ( F `
 suc  A )  /\  -.  v  e.  ( F `  A ) )  ->  ( F `  A )  =/=  ( F `  suc  A ) )
3428, 33syl6 34 . . . . . 6  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
3534expd 437 . . . . 5  |-  ( A  e.  om  ->  (
v  e.  ( x 
\  ( F `  A ) )  -> 
( ( v  i^i  ( x  \  ( F `  A )
) )  =  (/)  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) ) )
3635rexlimdv 2912 . . . 4  |-  ( A  e.  om  ->  ( E. v  e.  (
x  \  ( F `  A ) ) ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( F `
 A )  =/=  ( F `  suc  A ) ) )
3713, 36syl5 33 . . 3  |-  ( A  e.  om  ->  (
( x  \  ( F `  A )
)  =/=  (/)  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
389, 37syld 45 . 2  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  ( F `
 suc  A )
) )
3938com12 32 1  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   {crab 2775   _Vcvv 3080    \ cdif 3433    i^i cin 3435    C_ wss 3436   (/)c0 3761   U.cuni 4219    |-> cmpt 4482    |` cres 4855   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:  inf3lem4  8145
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