Theorem cardprc 9999

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 10580 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9563 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 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → (card‘𝑥) = (card‘𝑦))
2		id 22	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	1, 2	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
4	3	cbvabv 2806	. . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦}
5	4	cardprclem 9998	. 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
6	5	nelir 3040	1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

Syntax hints:   = wceq 1540  {cab 2714   ∉ wnel 3037  Vcvv 3464  ‘cfv 6536  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-oi 9529  df-har 9576  df-card 9958

This theorem is referenced by:  alephprc  10118