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Theorem cardprc 7823
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 8392 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 7469 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 5687 . . . . 5  |-  ( x  =  y  ->  ( card `  x )  =  ( card `  y
) )
2 id 20 . . . . 5  |-  ( x  =  y  ->  x  =  y )
31, 2eqeq12d 2418 . . . 4  |-  ( x  =  y  ->  (
( card `  x )  =  x  <->  ( card `  y
)  =  y ) )
43cbvabv 2523 . . 3  |-  { x  |  ( card `  x
)  =  x }  =  { y  |  (
card `  y )  =  y }
54cardprclem 7822 . 2  |-  -.  {
x  |  ( card `  x )  =  x }  e.  _V
6 df-nel 2570 . 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 1649    e. wcel 1721   {cab 2390    e/ wnel 2568   _Vcvv 2916   ` cfv 5413   cardccrd 7778
This theorem is referenced by:  alephprc  7936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6508  df-recs 6592  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-oi 7435  df-har 7482  df-card 7782
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