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Theorem cardprc 7497
Description: The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 8065 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7143 to construct (effectively)  ( aleph `  suc  A ) from  ( aleph `  A
), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
Assertion
Ref Expression
cardprc  |-  { x  |  ( card `  x
)  =  x }  e/  _V

Proof of Theorem cardprc
StepHypRef Expression
1 fveq2 5377 . . . . 5  |-  ( x  =  y  ->  ( card `  x )  =  ( card `  y
) )
2 id 21 . . . . 5  |-  ( x  =  y  ->  x  =  y )
31, 2eqeq12d 2267 . . . 4  |-  ( x  =  y  ->  (
( card `  x )  =  x  <->  ( card `  y
)  =  y ) )
43cbvabv 2368 . . 3  |-  { x  |  ( card `  x
)  =  x }  =  { y  |  (
card `  y )  =  y }
54cardprclem 7496 . 2  |-  -.  {
x  |  ( card `  x )  =  x }  e.  _V
6 df-nel 2415 . 2  |-  ( { x  |  ( card `  x )  =  x }  e/  _V  <->  -.  { x  |  ( card `  x
)  =  x }  e.  _V )
75, 6mpbir 202 1  |-  { x  |  ( card `  x
)  =  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    = wceq 1619    e. wcel 1621   {cab 2239    e/ wnel 2413   _Vcvv 2727   ` cfv 4592   cardccrd 7452
This theorem is referenced by:  alephprc  7610
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-oi 7109  df-har 7156  df-card 7456
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