Theorem cardprc 10085

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 10669 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9562 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 6905	. . . . 5 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2		id 23	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	1, 2	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
4	3	cbvabv 2834	. . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦}
5	4	cardprclem 10084	. 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
6	5	nelir 3068	1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

This proof depends on syntax axioms:   = wceq 1570  {cab 2741   ∉ wnel 3065  Vcvv 3454  ‘cfv 6555  cardccrd 10040

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-oi 9528  df-har 9575  df-card 10044

This theorem is used by:  alephprc  10202