MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inf3lem3 Structured version   Unicode version

Theorem inf3lem3 8046
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8051 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8020. (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 8043 . . . 4  |-  ( A  e.  om  ->  ( F `  A )  C_  x )
61, 2, 3, 4inf3lem2 8045 . . . . 5  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( A  e.  om  ->  ( F `  A
)  =/=  x ) )
76com12 31 . . . 4  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  x ) )
8 pssdifn0 3888 . . . 4  |-  ( ( ( F `  A
)  C_  x  /\  ( F `  A )  =/=  x )  -> 
( x  \  ( F `  A )
)  =/=  (/) )
95, 7, 8syl6an 545 . . 3  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( x  \  ( F `  A )
)  =/=  (/) ) )
10 vex 3116 . . . . . 6  |-  x  e. 
_V
11 difss 3631 . . . . . 6  |-  ( x 
\  ( F `  A ) )  C_  x
1210, 11ssexi 4592 . . . . 5  |-  ( x 
\  ( F `  A ) )  e. 
_V
1312zfreg 8020 . . . 4  |-  ( ( x  \  ( F `
 A ) )  =/=  (/)  ->  E. v  e.  ( x  \  ( F `  A )
) ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
14 eldifi 3626 . . . . . . . . . . 11  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  v  e.  x )
15 inssdif0 3894 . . . . . . . . . . . 12  |-  ( ( v  i^i  x ) 
C_  ( F `  A )  <->  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )
1615biimpri 206 . . . . . . . . . . 11  |-  ( ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( v  i^i  x )  C_  ( F `  A ) )
1714, 16anim12i 566 . . . . . . . . . 10  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  (
v  e.  x  /\  ( v  i^i  x
)  C_  ( F `  A ) ) )
18 vex 3116 . . . . . . . . . . 11  |-  v  e. 
_V
19 fvex 5875 . . . . . . . . . . 11  |-  ( F `
 A )  e. 
_V
201, 2, 18, 19inf3lema 8040 . . . . . . . . . 10  |-  ( v  e.  ( G `  ( F `  A ) )  <->  ( v  e.  x  /\  ( v  i^i  x )  C_  ( F `  A ) ) )
2117, 20sylibr 212 . . . . . . . . 9  |-  ( ( v  e.  ( x 
\  ( F `  A ) )  /\  ( v  i^i  (
x  \  ( F `  A ) ) )  =  (/) )  ->  v  e.  ( G `  ( F `  A )
) )
221, 2, 3, 4inf3lemc 8042 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( F `  suc  A )  =  ( G `  ( F `  A ) ) )
2322eleq2d 2537 . . . . . . . . 9  |-  ( A  e.  om  ->  (
v  e.  ( F `
 suc  A )  <->  v  e.  ( G `  ( F `  A ) ) ) )
2421, 23syl5ibr 221 . . . . . . . 8  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  v  e.  ( F `  suc  A
) ) )
25 eldifn 3627 . . . . . . . . . 10  |-  ( v  e.  ( x  \ 
( F `  A
) )  ->  -.  v  e.  ( F `  A ) )
2625adantr 465 . . . . . . . . 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 533 . . . . . . 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 2540 . . . . . . . . . 10  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 A )  <->  v  e.  ( F `  suc  A
) ) )
3029biimprd 223 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  (
v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) ) )
31 iman 424 . . . . . . . . 9  |-  ( ( v  e.  ( F `
 suc  A )  ->  v  e.  ( F `
 A ) )  <->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3230, 31sylib 196 . . . . . . . 8  |-  ( ( F `  A )  =  ( F `  suc  A )  ->  -.  ( v  e.  ( F `  suc  A
)  /\  -.  v  e.  ( F `  A
) ) )
3332necon2ai 2702 . . . . . . 7  |-  ( ( v  e.  ( F `
 suc  A )  /\  -.  v  e.  ( F `  A ) )  ->  ( F `  A )  =/=  ( F `  suc  A ) )
3428, 33syl6 33 . . . . . 6  |-  ( A  e.  om  ->  (
( v  e.  ( x  \  ( F `
 A ) )  /\  ( v  i^i  ( x  \  ( F `  A )
) )  =  (/) )  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
3534expd 436 . . . . 5  |-  ( A  e.  om  ->  (
v  e.  ( x 
\  ( F `  A ) )  -> 
( ( v  i^i  ( x  \  ( F `  A )
) )  =  (/)  ->  ( F `  A
)  =/=  ( F `
 suc  A )
) ) )
3635rexlimdv 2953 . . . 4  |-  ( A  e.  om  ->  ( E. v  e.  (
x  \  ( F `  A ) ) ( v  i^i  ( x 
\  ( F `  A ) ) )  =  (/)  ->  ( F `
 A )  =/=  ( F `  suc  A ) ) )
3713, 36syl5 32 . . 3  |-  ( A  e.  om  ->  (
( x  \  ( F `  A )
)  =/=  (/)  ->  ( F `  A )  =/=  ( F `  suc  A ) ) )
389, 37syld 44 . 2  |-  ( A  e.  om  ->  (
( x  =/=  (/)  /\  x  C_ 
U. x )  -> 
( F `  A
)  =/=  ( F `
 suc  A )
) )
3938com12 31 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   U.cuni 4245    |-> cmpt 4505   suc csuc 4880    |` cres 5001   ` cfv 5587   omcom 6679   reccrdg 7075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-reg 8017
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-om 6680  df-recs 7042  df-rdg 7076
This theorem is referenced by:  inf3lem4  8047
  Copyright terms: Public domain W3C validator