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Theorem cardprc 7856
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 8425 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7502 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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fveq2 5719 . . . . 5  |-  ( x  =  y  ->  ( card `  x )  =  ( card `  y
) )
2 id 20 . . . . 5  |-  ( x  =  y  ->  x  =  y )
31, 2eqeq12d 2449 . . . 4  |-  ( x  =  y  ->  (
( card `  x )  =  x  <->  ( card `  y
)  =  y ) )
43cbvabv 2554 . . 3  |-  { x  |  ( card `  x
)  =  x }  =  { y  |  (
card `  y )  =  y }
54cardprclem 7855 . 2  |-  -.  {
x  |  ( card `  x )  =  x }  e.  _V
6 df-nel 2601 . 2  |-  ( { x  |  ( card `  x )  =  x }  e/  _V  <->  -.  { x  |  ( card `  x
)  =  x }  e.  _V )
75, 6mpbir 201 1  |-  { x  |  ( card `  x
)  =  x }  e/  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   {cab 2421    e/ wnel 2599   _Vcvv 2948   ` cfv 5445   cardccrd 7811
This theorem is referenced by:  alephprc  7969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-oi 7468  df-har 7515  df-card 7815
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