Theorem cardprc 4861

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 4850	. . 3 |- (U.{x | (card` x) = x} e. V -> (card` U.{x | (card` x) = x}) e. (card` P~U.{x | (card` x) = x}))
2		fvex 3732	. . . . . . 7 |- (card` P~U.{x | (card` x) = x}) e. V
3		cardidm 4849	. . . . . . . . 9 |- (card` (card` P~U.{x | (card` x) = x})) = (card` P~U.{x | (card` x) = x})
4		ax-17 971	. . . . . . . . . . . 12 |- (y e. card -> A.x y e. card)
5		hbab1 1466	. . . . . . . . . . . . . 14 |- (y e. {x | (card` x) = x} -> A.x y e. {x | (card` x) = x})
6	5	hbuni 2509	. . . . . . . . . . . . 13 |- (y e. U.{x | (card` x) = x} -> A.x y e. U.{x | (card` x) = x})
7	6	hbpw 2407	. . . . . . . . . . . 12 |- (y e. P~U.{x | (card` x) = x} -> A.x y e. P~U.{x | (card` x) = x})
8	4, 7	hbfv 3729	. . . . . . . . . . 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 3729	. . . . . . . . . . . 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 1565	. . . . . . . . . . 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 3724	. . . . . . . . . . . 12 |- (x = (card` P~U.{x | (card` x) = x}) -> (card` x) = (card` (card` P~U.{x | (card` x) = x})))
12		id 59	. . . . . . . . . . . 12 |- (x = (card` P~U.{x | (card` x) = x}) -> x = (card` P~U.{x | (card` x) = x}))
13	11, 12	eqeq12d 1489	. . . . . . . . . . 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 1898	. . . . . . . . . 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 7	. . . . . . . . 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 190	. . . . . . . 8 |- (card` P~U.{x | (card` x) = x}) e. {x | (card` x) = x}
17		elssuni 2526	. . . . . . . 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 7	. . . . . . 7 |- (card` P~U.{x | (card` x) = x}) (_ U.{x | (card` x) = x}
19		ssdomg 4408	. . . . . . 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 4836	. . . . . . 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 695	. . . . . 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 194	. . . . 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 2106	. . . 4 |- (U.{x | (card` x) = x} e. V -> (card` P~U.{x | (card` x) = x}) (_ (card` U.{x | (card` x) = x}))
25		cardon 4827	. . . . 5 |- (card` P~U.{x | (card` x) = x}) e. On
26		cardon 4827	. . . . 5 |- (card` U.{x | (card` x) = x}) e. On
27		ontri1 2981	. . . . 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 697	. . . 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 198	. . 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 135	. 2 |- -. U.{x | (card` x) = x} e. V
31		uniexg 2871	. 2 |- ({x | (card` x) = x} e. V -> U.{x | (card` x) = x} e. V)
32	30, 31	mto 106	1 |- -. {x | (card` x) = x} e. V

Syntax hints:  -. wn 2   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~<_ cdom 4365  cardccrd 4813

This theorem is referenced by:  alephprc 4893

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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