Theorem cardprc 9939

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 10519 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9493 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.

Step	Hyp	Ref	Expression
1		fveq2 6868	. . . . 5 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	1, 2	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
4	3	cbvabv 2833	. . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦}
5	4	cardprclem 9938	. 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
6	5	nelir 3065	1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

Syntax hints:   = wceq 1561  {cab 2741   ∉ wnel 3062  Vcvv 3455  ‘cfv 6522  cardccrd 9894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-oi 9459  df-har 9506  df-card 9898

This theorem is referenced by:  alephprc  10056