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} e. V

Proof of Theorem cardprc

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

Syntax hints:  -. wn 1   <-> wb 127  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  P~cpw 1798  U.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

metamath.org