Theorem cardprc 4841

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 4830	. . 3 |- (U.{x | (card` x) = x} e. V -> (card` U.{x | (card` x) = x}) e. (card` P~U.{x | (card` x) = x}))
2		fvex 3723	. . . . . . 7 |- (card` P~U.{x | (card` x) = x}) e. V
3		cardidm 4829	. . . . . . . . 9 |- (card` (card` P~U.{x | (card` x) = x})) = (card` P~U.{x | (card` x) = x})
4		ax-17 969	. . . . . . . . . . . 12 |- (y e. card -> A.x y e. card)
5		hbab1 1464	. . . . . . . . . . . . . 14 |- (y e. {x | (card` x) = x} -> A.x y e. {x | (card` x) = x})
6	5	hbuni 2504	. . . . . . . . . . . . 13 |- (y e. U.{x | (card` x) = x} -> A.x y e. U.{x | (card` x) = x})
7	6	hbpw 2403	. . . . . . . . . . . 12 |- (y e. P~U.{x | (card` x) = x} -> A.x y e. P~U.{x | (card` x) = x})
8	4, 7	hbfv 3720	. . . . . . . . . . 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 3720	. . . . . . . . . . . 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 1562	. . . . . . . . . . 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 3715	. . . . . . . . . . . 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 1486	. . . . . . . . . . 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 1894	. . . . . . . . . 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 2521	. . . . . . . 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 4395	. . . . . . 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 4816	. . . . . . 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 694	. . . . . 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 2102	. . . 4 |- (U.{x | (card` x) = x} e. V -> (card` P~U.{x | (card` x) = x}) (_ (card` U.{x | (card` x) = x}))
25		cardon 4807	. . . . 5 |- (card` P~U.{x | (card` x) = x}) e. On
26		cardon 4807	. . . . 5 |- (card` U.{x | (card` x) = x}) e. On
27		ontri1 2976	. . . . 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 696	. . . 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 2866	. 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 954   e. wcel 956  {cab 1461  Vcvv 1807   (_ wss 2043  P~cpw 2397  U.cuni 2498   class class class wbr 2614  Oncon0 2943  ` cfv 3177   ~<_ cdom 4355  cardccrd 4793

This theorem is referenced by:  alephprc 4873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796

Copyright terms: Public domain