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Theorem cardprc 9401
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 9975 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9000 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.)
Assertion
Ref Expression
cardprc {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

Proof of Theorem cardprc
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6666 . . . . 5 (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
31, 2eqeq12d 2841 . . . 4 (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
43cbvabv 2893 . . 3 {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦}
54cardprclem 9400 . 2 ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
65nelir 3130 1 {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  {cab 2803  wnel 3127  Vcvv 3499  cfv 6351  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-wrecs 7941  df-recs 8002  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-oi 8966  df-har 9014  df-card 9360
This theorem is referenced by:  alephprc  9517
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