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Theorem cardprclem 8359
Description: Lemma for cardprc 8360. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
cardprclem.1  |-  A  =  { x  |  (
card `  x )  =  x }
Assertion
Ref Expression
cardprclem  |-  -.  A  e.  _V
Distinct variable group:    x, A

Proof of Theorem cardprclem
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardprclem.1 . . . . . . . . 9  |-  A  =  { x  |  (
card `  x )  =  x }
21eleq2i 2545 . . . . . . . 8  |-  ( x  e.  A  <->  x  e.  { x  |  ( card `  x )  =  x } )
3 abid 2454 . . . . . . . 8  |-  ( x  e.  { x  |  ( card `  x
)  =  x }  <->  (
card `  x )  =  x )
4 iscard 8355 . . . . . . . 8  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
52, 3, 43bitri 271 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
65simplbi 460 . . . . . 6  |-  ( x  e.  A  ->  x  e.  On )
76ssriv 3508 . . . . 5  |-  A  C_  On
8 ssonuni 6601 . . . . 5  |-  ( A  e.  _V  ->  ( A  C_  On  ->  U. A  e.  On ) )
97, 8mpi 17 . . . 4  |-  ( A  e.  _V  ->  U. A  e.  On )
10 domrefg 7550 . . . . 5  |-  ( U. A  e.  On  ->  U. A  ~<_  U. A )
119, 10syl 16 . . . 4  |-  ( A  e.  _V  ->  U. A  ~<_  U. A )
12 elharval 7988 . . . 4  |-  ( U. A  e.  (har `  U. A )  <->  ( U. A  e.  On  /\  U. A  ~<_  U. A ) )
139, 11, 12sylanbrc 664 . . 3  |-  ( A  e.  _V  ->  U. A  e.  (har `  U. A ) )
147sseli 3500 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  On )
15 domrefg 7550 . . . . . . . . . 10  |-  ( z  e.  On  ->  z  ~<_  z )
1615ancli 551 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  e.  On  /\  z  ~<_  z ) )
17 elharval 7988 . . . . . . . . 9  |-  ( z  e.  (har `  z
)  <->  ( z  e.  On  /\  z  ~<_  z ) )
1816, 17sylibr 212 . . . . . . . 8  |-  ( z  e.  On  ->  z  e.  (har `  z )
)
1914, 18syl 16 . . . . . . 7  |-  ( z  e.  A  ->  z  e.  (har `  z )
)
20 harcard 8358 . . . . . . . 8  |-  ( card `  (har `  z )
)  =  (har `  z )
21 fvex 5875 . . . . . . . . 9  |-  (har `  z )  e.  _V
22 fveq2 5865 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  ( card `  x )  =  (
card `  (har `  z
) ) )
23 id 22 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  x  =  (har `  z ) )
2422, 23eqeq12d 2489 . . . . . . . . 9  |-  ( x  =  (har `  z
)  ->  ( ( card `  x )  =  x  <->  ( card `  (har `  z ) )  =  (har `  z )
) )
2521, 24, 1elab2 3253 . . . . . . . 8  |-  ( (har
`  z )  e.  A  <->  ( card `  (har `  z ) )  =  (har `  z )
)
2620, 25mpbir 209 . . . . . . 7  |-  (har `  z )  e.  A
27 eleq2 2540 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( z  e.  w  <->  z  e.  (har
`  z ) ) )
28 eleq1 2539 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( w  e.  A  <->  (har `  z )  e.  A ) )
2927, 28anbi12d 710 . . . . . . . 8  |-  ( w  =  (har `  z
)  ->  ( (
z  e.  w  /\  w  e.  A )  <->  ( z  e.  (har `  z )  /\  (har `  z )  e.  A
) ) )
3021, 29spcev 3205 . . . . . . 7  |-  ( ( z  e.  (har `  z )  /\  (har `  z )  e.  A
)  ->  E. w
( z  e.  w  /\  w  e.  A
) )
3119, 26, 30sylancl 662 . . . . . 6  |-  ( z  e.  A  ->  E. w
( z  e.  w  /\  w  e.  A
) )
32 eluni 4248 . . . . . 6  |-  ( z  e.  U. A  <->  E. w
( z  e.  w  /\  w  e.  A
) )
3331, 32sylibr 212 . . . . 5  |-  ( z  e.  A  ->  z  e.  U. A )
3433ssriv 3508 . . . 4  |-  A  C_  U. A
35 harcard 8358 . . . . 5  |-  ( card `  (har `  U. A ) )  =  (har `  U. A )
36 fvex 5875 . . . . . 6  |-  (har `  U. A )  e.  _V
37 fveq2 5865 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  ( card `  x )  =  ( card `  (har ` 
U. A ) ) )
38 id 22 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  x  =  (har `  U. A ) )
3937, 38eqeq12d 2489 . . . . . 6  |-  ( x  =  (har `  U. A )  ->  (
( card `  x )  =  x  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) ) )
4036, 39, 1elab2 3253 . . . . 5  |-  ( (har
`  U. A )  e.  A  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) )
4135, 40mpbir 209 . . . 4  |-  (har `  U. A )  e.  A
4234, 41sselii 3501 . . 3  |-  (har `  U. A )  e.  U. A
4313, 42jctir 538 . 2  |-  ( A  e.  _V  ->  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
44 eloni 4888 . . 3  |-  ( U. A  e.  On  ->  Ord  U. A )
45 ordn2lp 4898 . . 3  |-  ( Ord  U. A  ->  -.  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
469, 44, 453syl 20 . 2  |-  ( A  e.  _V  ->  -.  ( U. A  e.  (har
`  U. A )  /\  (har `  U. A )  e.  U. A ) )
4743, 46pm2.65i 173 1  |-  -.  A  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113    C_ wss 3476   U.cuni 4245   class class class wbr 4447   Ord word 4877   Oncon0 4878   ` cfv 5587    ~<_ cdom 7514    ~< csdm 7515  harchar 7981   cardccrd 8315
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
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-rmo 2822  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-int 4283  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-se 4839  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-isom 5596  df-riota 6244  df-recs 7042  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-oi 7934  df-har 7983  df-card 8319
This theorem is referenced by:  cardprc  8360
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