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Theorem cardprc 7609
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 8179 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7255 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
Dummy variable  y is distinct from all other variables.

Proof of Theorem cardprc
StepHypRef Expression
1 fveq2 5486 . . . . 5  |-  ( x  =  y  ->  ( card `  x )  =  ( card `  y
) )
2 id 21 . . . . 5  |-  ( x  =  y  ->  x  =  y )
31, 2eqeq12d 2299 . . . 4  |-  ( x  =  y  ->  (
( card `  x )  =  x  <->  ( card `  y
)  =  y ) )
43cbvabv 2404 . . 3  |-  { x  |  ( card `  x
)  =  x }  =  { y  |  (
card `  y )  =  y }
54cardprclem 7608 . 2  |-  -.  {
x  |  ( card `  x )  =  x }  e.  _V
6 df-nel 2451 . 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 1624    e. wcel 1685   {cab 2271    e/ wnel 2449   _Vcvv 2790   ` cfv 5222   cardccrd 7564
This theorem is referenced by:  alephprc  7722
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-iota 6253  df-riota 6300  df-recs 6384  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-oi 7221  df-har 7268  df-card 7568
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