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Theorem cardprclem 8431
Description: Lemma for cardprc 8432. (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 2541 . . . . . . . 8  |-  ( x  e.  A  <->  x  e.  { x  |  ( card `  x )  =  x } )
3 abid 2459 . . . . . . . 8  |-  ( x  e.  { x  |  ( card `  x
)  =  x }  <->  (
card `  x )  =  x )
4 iscard 8427 . . . . . . . 8  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
52, 3, 43bitri 279 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
65simplbi 467 . . . . . 6  |-  ( x  e.  A  ->  x  e.  On )
76ssriv 3422 . . . . 5  |-  A  C_  On
8 ssonuni 6632 . . . . 5  |-  ( A  e.  _V  ->  ( A  C_  On  ->  U. A  e.  On ) )
97, 8mpi 20 . . . 4  |-  ( A  e.  _V  ->  U. A  e.  On )
10 domrefg 7622 . . . . 5  |-  ( U. A  e.  On  ->  U. A  ~<_  U. A )
119, 10syl 17 . . . 4  |-  ( A  e.  _V  ->  U. A  ~<_  U. A )
12 elharval 8096 . . . 4  |-  ( U. A  e.  (har `  U. A )  <->  ( U. A  e.  On  /\  U. A  ~<_  U. A ) )
139, 11, 12sylanbrc 677 . . 3  |-  ( A  e.  _V  ->  U. A  e.  (har `  U. A ) )
147sseli 3414 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  On )
15 domrefg 7622 . . . . . . . . . 10  |-  ( z  e.  On  ->  z  ~<_  z )
1615ancli 560 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  e.  On  /\  z  ~<_  z ) )
17 elharval 8096 . . . . . . . . 9  |-  ( z  e.  (har `  z
)  <->  ( z  e.  On  /\  z  ~<_  z ) )
1816, 17sylibr 217 . . . . . . . 8  |-  ( z  e.  On  ->  z  e.  (har `  z )
)
1914, 18syl 17 . . . . . . 7  |-  ( z  e.  A  ->  z  e.  (har `  z )
)
20 harcard 8430 . . . . . . . 8  |-  ( card `  (har `  z )
)  =  (har `  z )
21 fvex 5889 . . . . . . . . 9  |-  (har `  z )  e.  _V
22 fveq2 5879 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  ( card `  x )  =  (
card `  (har `  z
) ) )
23 id 22 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  x  =  (har `  z ) )
2422, 23eqeq12d 2486 . . . . . . . . 9  |-  ( x  =  (har `  z
)  ->  ( ( card `  x )  =  x  <->  ( card `  (har `  z ) )  =  (har `  z )
) )
2521, 24, 1elab2 3176 . . . . . . . 8  |-  ( (har
`  z )  e.  A  <->  ( card `  (har `  z ) )  =  (har `  z )
)
2620, 25mpbir 214 . . . . . . 7  |-  (har `  z )  e.  A
27 eleq2 2538 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( z  e.  w  <->  z  e.  (har
`  z ) ) )
28 eleq1 2537 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( w  e.  A  <->  (har `  z )  e.  A ) )
2927, 28anbi12d 725 . . . . . . . 8  |-  ( w  =  (har `  z
)  ->  ( (
z  e.  w  /\  w  e.  A )  <->  ( z  e.  (har `  z )  /\  (har `  z )  e.  A
) ) )
3021, 29spcev 3127 . . . . . . 7  |-  ( ( z  e.  (har `  z )  /\  (har `  z )  e.  A
)  ->  E. w
( z  e.  w  /\  w  e.  A
) )
3119, 26, 30sylancl 675 . . . . . 6  |-  ( z  e.  A  ->  E. w
( z  e.  w  /\  w  e.  A
) )
32 eluni 4193 . . . . . 6  |-  ( z  e.  U. A  <->  E. w
( z  e.  w  /\  w  e.  A
) )
3331, 32sylibr 217 . . . . 5  |-  ( z  e.  A  ->  z  e.  U. A )
3433ssriv 3422 . . . 4  |-  A  C_  U. A
35 harcard 8430 . . . . 5  |-  ( card `  (har `  U. A ) )  =  (har `  U. A )
36 fvex 5889 . . . . . 6  |-  (har `  U. A )  e.  _V
37 fveq2 5879 . . . . . . 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 2486 . . . . . 6  |-  ( x  =  (har `  U. A )  ->  (
( card `  x )  =  x  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) ) )
4036, 39, 1elab2 3176 . . . . 5  |-  ( (har
`  U. A )  e.  A  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) )
4135, 40mpbir 214 . . . 4  |-  (har `  U. A )  e.  A
4234, 41sselii 3415 . . 3  |-  (har `  U. A )  e.  U. A
4313, 42jctir 547 . 2  |-  ( A  e.  _V  ->  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
44 eloni 5440 . . 3  |-  ( U. A  e.  On  ->  Ord  U. A )
45 ordn2lp 5450 . . 3  |-  ( Ord  U. A  ->  -.  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
469, 44, 453syl 18 . 2  |-  ( A  e.  _V  ->  -.  ( U. A  e.  (har
`  U. A )  /\  (har `  U. A )  e.  U. A ) )
4743, 46pm2.65i 178 1  |-  -.  A  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   A.wral 2756   _Vcvv 3031    C_ wss 3390   U.cuni 4190   class class class wbr 4395   Ord word 5429   Oncon0 5430   ` cfv 5589    ~<_ cdom 7585    ~< csdm 7586  harchar 8089   cardccrd 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-card 8391
This theorem is referenced by:  cardprc  8432
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