Theorem cardprc 10010

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 10594 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9523 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 23	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
3	1, 2	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝑦 → ((card‘𝑥) = 𝑥 ↔ (card‘𝑦) = 𝑦))
4	3	cbvabv 2830	. . 3 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} = {𝑦 ∣ (card‘𝑦) = 𝑦}
5	4	cardprclem 10009	. 2 ⊢ ¬ {𝑥 ∣ (card‘𝑥) = 𝑥} ∈ V
6	5	nelir 3064	1 ⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V

This proof depends on syntax axioms:   = wceq 1570  {cab 2738   ∉ wnel 3061  Vcvv 3450  ‘cfv 6535  cardccrd 9965

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-er 8703  df-en 8960  df-dom 8961  df-sdom 8962  df-oi 9489  df-har 9536  df-card 9969

This theorem is used by:  alephprc  10127