Theorem cardprc 3667

Description: The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310.

Assertion

Ref	Expression
cardprc	⊢ ¬ {x∣(card ‘x) = x} ∈ V

Proof of Theorem cardprc

Step	Hyp	Ref	Expression
1		canth3 3656	. . 3 ⊢ (∪{x∣(card ‘x) = x} ∈ V → (card ‘∪{x∣(card ‘x) = x}) ∈ (card ‘℘∪{x∣(card ‘x) = x}))
2		fvex 2838	. . . . . . 7 ⊢ (card ‘℘∪{x∣(card ‘x) = x}) ∈ V
3		cardcard 3655	. . . . . . . . 9 ⊢ (card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x})
4		ax-17 925	. . . . . . . . . . . 12 ⊢ (y ∈ card → ∀x y ∈ card)
5		hbab1 1095	. . . . . . . . . . . . . 14 ⊢ (y ∈ {x∣(card ‘x) = x} → ∀x y ∈ {x∣(card ‘x) = x})
6	5	hbuni 1925	. . . . . . . . . . . . 13 ⊢ (y ∈ ∪{x∣(card ‘x) = x} → ∀x y ∈ ∪{x∣(card ‘x) = x})
7	6	hbpw 1804	. . . . . . . . . . . 12 ⊢ (y ∈ ℘∪{x∣(card ‘x) = x} → ∀x y ∈ ℘∪{x∣(card ‘x) = x})
8	4, 7	hbfv 2837	. . . . . . . . . . 11 ⊢ (y ∈ (card ‘℘∪{x∣(card ‘x) = x}) → ∀x y ∈ (card ‘℘∪{x∣(card ‘x) = x}))
9	4, 8	hbfv 2837	. . . . . . . . . . . 12 ⊢ (y ∈ (card ‘(card ‘℘∪{x∣(card ‘x) = x})) → ∀x y ∈ (card ‘(card ‘℘∪{x∣(card ‘x) = x})))
10	9, 8	hbeq 1171	. . . . . . . . . . 11 ⊢ ((card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x}) → ∀x(card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x}))
11		fveq2 2832	. . . . . . . . . . . 12 ⊢ (x = (card ‘℘∪{x∣(card ‘x) = x}) → (card ‘x) = (card ‘(card ‘℘∪{x∣(card ‘x) = x})))
12		id 9	. . . . . . . . . . . 12 ⊢ (x = (card ‘℘∪{x∣(card ‘x) = x}) → x = (card ‘℘∪{x∣(card ‘x) = x}))
13	11, 12	cleq12d 1115	. . . . . . . . . . 11 ⊢ (x = (card ‘℘∪{x∣(card ‘x) = x}) → ((card ‘x) = x ↔ (card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x})))
14	8, 10, 13	elabgf 1416	. . . . . . . . . 10 ⊢ ((card ‘℘∪{x∣(card ‘x) = x}) ∈ V → ((card ‘℘∪{x∣(card ‘x) = x}) ∈ {x∣(card ‘x) = x} ↔ (card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x})))
15	2, 14	ax-mp 6	. . . . . . . . 9 ⊢ ((card ‘℘∪{x∣(card ‘x) = x}) ∈ {x∣(card ‘x) = x} ↔ (card ‘(card ‘℘∪{x∣(card ‘x) = x})) = (card ‘℘∪{x∣(card ‘x) = x}))
16	3, 15	mpbir 165	. . . . . . . 8 ⊢ (card ‘℘∪{x∣(card ‘x) = x}) ∈ {x∣(card ‘x) = x}
17		elssuni 1940	. . . . . . . 8 ⊢ ((card ‘℘∪{x∣(card ‘x) = x}) ∈ {x∣(card ‘x) = x} → (card ‘℘∪{x∣(card ‘x) = x}) ⊆ ∪{x∣(card ‘x) = x})
18	16, 17	ax-mp 6	. . . . . . 7 ⊢ (card ‘℘∪{x∣(card ‘x) = x}) ⊆ ∪{x∣(card ‘x) = x}
19		ssdomg 3311	. . . . . . 7 ⊢ ((card ‘℘∪{x∣(card ‘x) = x}) ∈ V → ((card ‘℘∪{x∣(card ‘x) = x}) ⊆ ∪{x∣(card ‘x) = x} → (card ‘℘∪{x∣(card ‘x) = x}) ≼ ∪{x∣(card ‘x) = x}))
20	2, 18, 19	mp2 43	. . . . . 6 ⊢ (card ‘℘∪{x∣(card ‘x) = x}) ≼ ∪{x∣(card ‘x) = x}
21		carddom 3642	. . . . . . 7 ⊢ (((card ‘℘∪{x∣(card ‘x) = x}) ∈ V ∧ ∪{x∣(card ‘x) = x} ∈ V) → ((card ‘(card ‘℘∪{x∣(card ‘x) = x})) ⊆ (card ‘∪{x∣(card ‘x) = x}) ↔ (card ‘℘∪{x∣(card ‘x) = x}) ≼ ∪{x∣(card ‘x) = x}))
22	2, 21	mpan 518	. . . . . 6 ⊢ (∪{x∣(card ‘x) = x} ∈ V → ((card ‘(card ‘℘∪{x∣(card ‘x) = x})) ⊆ (card ‘∪{x∣(card ‘x) = x}) ↔ (card ‘℘∪{x∣(card ‘x) = x}) ≼ ∪{x∣(card ‘x) = x}))
23	20, 22	mpbiri 169	. . . . 5 ⊢ (∪{x∣(card ‘x) = x} ∈ V → (card ‘(card ‘℘∪{x∣(card ‘x) = x})) ⊆ (card ‘∪{x∣(card ‘x) = x}))
24	23, 3	syl5ssr 1545	. . . 4 ⊢ (∪{x∣(card ‘x) = x} ∈ V → (card ‘℘∪{x∣(card ‘x) = x}) ⊆ (card ‘∪{x∣(card ‘x) = x}))
25		cardon 3634	. . . . 5 ⊢ (card ‘℘∪{x∣(card ‘x) = x}) ∈ On
26		cardon 3634	. . . . 5 ⊢ (card ‘∪{x∣(card ‘x) = x}) ∈ On
27		ontri1 2232	. . . . 5 ⊢ (((card ‘℘∪{x∣(card ‘x) = x}) ∈ On ∧ (card ‘∪{x∣(card ‘x) = x}) ∈ On) → ((card ‘℘∪{x∣(card ‘x) = x}) ⊆ (card ‘∪{x∣(card ‘x) = x}) ↔ ¬ (card ‘∪{x∣(card ‘x) = x}) ∈ (card ‘℘∪{x∣(card ‘x) = x})))
28	25, 26, 27	mp2an 520	. . . 4 ⊢ ((card ‘℘∪{x∣(card ‘x) = x}) ⊆ (card ‘∪{x∣(card ‘x) = x}) ↔ ¬ (card ‘∪{x∣(card ‘x) = x}) ∈ (card ‘℘∪{x∣(card ‘x) = x}))
29	24, 28	sylib 173	. . 3 ⊢ (∪{x∣(card ‘x) = x} ∈ V → ¬ (card ‘∪{x∣(card ‘x) = x}) ∈ (card ‘℘∪{x∣(card ‘x) = x}))
30	1, 29	pm2.65i 116	. 2 ⊢ ¬ ∪{x∣(card ‘x) = x} ∈ V
31		uniexg 1948	. 2 ⊢ ({x∣(card ‘x) = x} ∈ V → ∪{x∣(card ‘x) = x} ∈ V)
32	30, 31	mto 93	1 ⊢ ¬ {x∣(card ‘x) = x} ∈ V

Syntax hints:  ¬ wn 1   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  ∪cuni 1919   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≼ cdom 3272  cardccrd 3620

This theorem is referenced by:  alephprc 3698

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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