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Theorem cardprclem 7702
Description: Lemma for cardprc 7703. (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 2422 . . . . . . . 8  |-  ( x  e.  A  <->  x  e.  { x  |  ( card `  x )  =  x } )
3 abid 2346 . . . . . . . 8  |-  ( x  e.  { x  |  ( card `  x
)  =  x }  <->  (
card `  x )  =  x )
4 iscard 7698 . . . . . . . 8  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
52, 3, 43bitri 262 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
65simplbi 446 . . . . . 6  |-  ( x  e.  A  ->  x  e.  On )
76ssriv 3260 . . . . 5  |-  A  C_  On
8 ssonuni 4660 . . . . 5  |-  ( A  e.  _V  ->  ( A  C_  On  ->  U. A  e.  On ) )
97, 8mpi 16 . . . 4  |-  ( A  e.  _V  ->  U. A  e.  On )
10 domrefg 6984 . . . . 5  |-  ( U. A  e.  On  ->  U. A  ~<_  U. A )
119, 10syl 15 . . . 4  |-  ( A  e.  _V  ->  U. A  ~<_  U. A )
12 elharval 7367 . . . 4  |-  ( U. A  e.  (har `  U. A )  <->  ( U. A  e.  On  /\  U. A  ~<_  U. A ) )
139, 11, 12sylanbrc 645 . . 3  |-  ( A  e.  _V  ->  U. A  e.  (har `  U. A ) )
147sseli 3252 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  On )
15 domrefg 6984 . . . . . . . . . 10  |-  ( z  e.  On  ->  z  ~<_  z )
1615ancli 534 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  e.  On  /\  z  ~<_  z ) )
17 elharval 7367 . . . . . . . . 9  |-  ( z  e.  (har `  z
)  <->  ( z  e.  On  /\  z  ~<_  z ) )
1816, 17sylibr 203 . . . . . . . 8  |-  ( z  e.  On  ->  z  e.  (har `  z )
)
1914, 18syl 15 . . . . . . 7  |-  ( z  e.  A  ->  z  e.  (har `  z )
)
20 harcard 7701 . . . . . . . 8  |-  ( card `  (har `  z )
)  =  (har `  z )
21 fvex 5622 . . . . . . . . 9  |-  (har `  z )  e.  _V
22 fveq2 5608 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  ( card `  x )  =  (
card `  (har `  z
) ) )
23 id 19 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  x  =  (har `  z ) )
2422, 23eqeq12d 2372 . . . . . . . . 9  |-  ( x  =  (har `  z
)  ->  ( ( card `  x )  =  x  <->  ( card `  (har `  z ) )  =  (har `  z )
) )
2521, 24, 1elab2 2993 . . . . . . . 8  |-  ( (har
`  z )  e.  A  <->  ( card `  (har `  z ) )  =  (har `  z )
)
2620, 25mpbir 200 . . . . . . 7  |-  (har `  z )  e.  A
27 eleq2 2419 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( z  e.  w  <->  z  e.  (har
`  z ) ) )
28 eleq1 2418 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( w  e.  A  <->  (har `  z )  e.  A ) )
2927, 28anbi12d 691 . . . . . . . 8  |-  ( w  =  (har `  z
)  ->  ( (
z  e.  w  /\  w  e.  A )  <->  ( z  e.  (har `  z )  /\  (har `  z )  e.  A
) ) )
3021, 29spcev 2951 . . . . . . 7  |-  ( ( z  e.  (har `  z )  /\  (har `  z )  e.  A
)  ->  E. w
( z  e.  w  /\  w  e.  A
) )
3119, 26, 30sylancl 643 . . . . . 6  |-  ( z  e.  A  ->  E. w
( z  e.  w  /\  w  e.  A
) )
32 eluni 3911 . . . . . 6  |-  ( z  e.  U. A  <->  E. w
( z  e.  w  /\  w  e.  A
) )
3331, 32sylibr 203 . . . . 5  |-  ( z  e.  A  ->  z  e.  U. A )
3433ssriv 3260 . . . 4  |-  A  C_  U. A
35 harcard 7701 . . . . 5  |-  ( card `  (har `  U. A ) )  =  (har `  U. A )
36 fvex 5622 . . . . . 6  |-  (har `  U. A )  e.  _V
37 fveq2 5608 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  ( card `  x )  =  ( card `  (har ` 
U. A ) ) )
38 id 19 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  x  =  (har `  U. A ) )
3937, 38eqeq12d 2372 . . . . . 6  |-  ( x  =  (har `  U. A )  ->  (
( card `  x )  =  x  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) ) )
4036, 39, 1elab2 2993 . . . . 5  |-  ( (har
`  U. A )  e.  A  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) )
4135, 40mpbir 200 . . . 4  |-  (har `  U. A )  e.  A
4234, 41sselii 3253 . . 3  |-  (har `  U. A )  e.  U. A
4313, 42jctir 524 . 2  |-  ( A  e.  _V  ->  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
44 eloni 4484 . . 3  |-  ( U. A  e.  On  ->  Ord  U. A )
45 ordn2lp 4494 . . 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 165 1  |-  -.  A  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   _Vcvv 2864    C_ wss 3228   U.cuni 3908   class class class wbr 4104   Ord word 4473   Oncon0 4474   ` cfv 5337    ~<_ cdom 6949    ~< csdm 6950  harchar 7360   cardccrd 7658
This theorem is referenced by:  cardprc  7703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-recs 6475  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-oi 7315  df-har 7362  df-card 7662
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