Theorem cardprc 6013

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 6002	. . 3 |- (U.{x | (card` x) = x} e. _V -> (card` U.{x | (card` x) = x}) e. (card` ~PU.{x | (card` x) = x}))
2		fvex 4689	. . . . . . 7 |- (card` ~PU.{x | (card` x) = x}) e. _V
3		cardidm 6001	. . . . . . . . 9 |- (card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x})
4		ax-17 1317	. . . . . . . . . . . 12 |- (y e. card -> A.x y e. card)
5		hbab1 1874	. . . . . . . . . . . . . 14 |- (y e. {x | (card` x) = x} -> A.x y e. {x | (card` x) = x})
6	5	hbuni 3183	. . . . . . . . . . . . 13 |- (y e. U.{x | (card` x) = x} -> A.x y e. U.{x | (card` x) = x})
7	6	hbpw 3041	. . . . . . . . . . . 12 |- (y e. ~PU.{x | (card` x) = x} -> A.x y e. ~PU.{x | (card` x) = x})
8	4, 7	hbfv 4686	. . . . . . . . . . 11 |- (y e. (card` ~PU.{x | (card` x) = x}) -> A.x y e. (card` ~PU.{x | (card` x) = x}))
9	4, 8	hbfv 4686	. . . . . . . . . . . 12 |- (y e. (card` (card` ~PU.{x | (card` x) = x})) -> A.x y e. (card` (card` ~PU.{x | (card` x) = x})))
10	9, 8	hbeq 1995	. . . . . . . . . . 11 |- ((card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x}) -> A.x(card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x}))
11		fveq2 4681	. . . . . . . . . . . 12 |- (x = (card` ~PU.{x | (card` x) = x}) -> (card` x) = (card` (card` ~PU.{x | (card` x) = x})))
12		id 73	. . . . . . . . . . . 12 |- (x = (card` ~PU.{x | (card` x) = x}) -> x = (card` ~PU.{x | (card` x) = x}))
13	11, 12	eqeq12d 1899	. . . . . . . . . . 11 |- (x = (card` ~PU.{x | (card` x) = x}) -> ((card` x) = x <-> (card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x})))
14	8, 10, 13	elabgf 2404	. . . . . . . . . 10 |- ((card` ~PU.{x | (card` x) = x}) e. _V -> ((card` ~PU.{x | (card` x) = x}) e. {x | (card` x) = x} <-> (card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x})))
15	2, 14	ax-mp 7	. . . . . . . . 9 |- ((card` ~PU.{x | (card` x) = x}) e. {x | (card` x) = x} <-> (card` (card` ~PU.{x | (card` x) = x})) = (card` ~PU.{x | (card` x) = x}))
16	3, 15	mpbir 207	. . . . . . . 8 |- (card` ~PU.{x | (card` x) = x}) e. {x | (card` x) = x}
17		elssuni 3206	. . . . . . . 8 |- ((card` ~PU.{x | (card` x) = x}) e. {x | (card` x) = x} -> (card` ~PU.{x | (card` x) = x}) C_ U.{x | (card` x) = x})
18	16, 17	ax-mp 7	. . . . . . 7 |- (card` ~PU.{x | (card` x) = x}) C_ U.{x | (card` x) = x}
19		ssdomg 5467	. . . . . . 7 |- ((card` ~PU.{x | (card` x) = x}) e. _V -> ((card` ~PU.{x | (card` x) = x}) C_ U.{x | (card` x) = x} -> (card` ~PU.{x | (card` x) = x}) ~<_ U.{x | (card` x) = x}))
20	2, 18, 19	mp2 54	. . . . . 6 |- (card` ~PU.{x | (card` x) = x}) ~<_ U.{x | (card` x) = x}
21		carddom 5987	. . . . . . 7 |- (((card` ~PU.{x | (card` x) = x}) e. _V /\ U.{x | (card` x) = x} e. _V) -> ((card` (card` ~PU.{x | (card` x) = x})) C_ (card` U.{x | (card` x) = x}) <-> (card` ~PU.{x | (card` x) = x}) ~<_ U.{x | (card` x) = x}))
22	2, 21	mpan 759	. . . . . 6 |- (U.{x | (card` x) = x} e. _V -> ((card` (card` ~PU.{x | (card` x) = x})) C_ (card` U.{x | (card` x) = x}) <-> (card` ~PU.{x | (card` x) = x}) ~<_ U.{x | (card` x) = x}))
23	20, 22	mpbiri 211	. . . . 5 |- (U.{x | (card` x) = x} e. _V -> (card` (card` ~PU.{x | (card` x) = x})) C_ (card` U.{x | (card` x) = x}))
24	23, 3	syl5ssr 2662	. . . 4 |- (U.{x | (card` x) = x} e. _V -> (card` ~PU.{x | (card` x) = x}) C_ (card` U.{x | (card` x) = x}))
25		cardon 5976	. . . . 5 |- (card` ~PU.{x | (card` x) = x}) e. On
26		cardon 5976	. . . . 5 |- (card` U.{x | (card` x) = x}) e. On
27		ontri1 3695	. . . . 5 |- (((card` ~PU.{x | (card` x) = x}) e. On /\ (card` U.{x | (card` x) = x}) e. On) -> ((card` ~PU.{x | (card` x) = x}) C_ (card` U.{x | (card` x) = x}) <-> -. (card` U.{x | (card` x) = x}) e. (card` ~PU.{x | (card` x) = x})))
28	25, 26, 27	mp2an 761	. . . 4 |- ((card` ~PU.{x | (card` x) = x}) C_ (card` U.{x | (card` x) = x}) <-> -. (card` U.{x | (card` x) = x}) e. (card` ~PU.{x | (card` x) = x}))
29	24, 28	sylib 215	. . 3 |- (U.{x | (card` x) = x} e. _V -> -. (card` U.{x | (card` x) = x}) e. (card` ~PU.{x | (card` x) = x}))
30	1, 29	pm2.65i 150	. 2 |- -. U.{x | (card` x) = x} e. _V
31		uniexg 3795	. 2 |- ({x | (card` x) = x} e. _V -> U.{x | (card` x) = x} e. _V)
32	30, 31	mto 121	1 |- -. {x | (card` x) = x} e. _V

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  U.cuni 3177   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~<_ cdom 5424  cardccrd 5859

This theorem is referenced by:  alephprc 6041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862

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