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Theorem cardprc 7567
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 8137 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7213 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 5444 . . . . 5  |-  ( x  =  y  ->  ( card `  x )  =  ( card `  y
) )
2 id 21 . . . . 5  |-  ( x  =  y  ->  x  =  y )
31, 2eqeq12d 2270 . . . 4  |-  ( x  =  y  ->  (
( card `  x )  =  x  <->  ( card `  y
)  =  y ) )
43cbvabv 2375 . . 3  |-  { x  |  ( card `  x
)  =  x }  =  { y  |  (
card `  y )  =  y }
54cardprclem 7566 . 2  |-  -.  {
x  |  ( card `  x )  =  x }  e.  _V
6 df-nel 2422 . 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 2242    e/ wnel 2420   _Vcvv 2757   ` cfv 4659   cardccrd 7522
This theorem is referenced by:  alephprc  7680
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-iota 6211  df-riota 6258  df-recs 6342  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-oi 7179  df-har 7226  df-card 7526
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