Theorem cardprclem 8359

Description: Lemma for cardprc 8360. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
cardprclem.1	|- A = { x | ( card ` x ) = x }

Assertion

Ref	Expression
cardprclem	|- -. A e. _V

Distinct variable group:   x, A

Proof of Theorem cardprclem

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardprclem.1	. . . . . . . . 9 |- A = { x | ( card ` x ) = x }
2	1	eleq2i 2545	. . . . . . . 8 |- ( x e. A <-> x e. { x | ( card ` x ) = x } )
3		abid 2454	. . . . . . . 8 |- ( x e. { x | ( card ` x ) = x } <-> ( card ` x ) = x )
4		iscard 8355	. . . . . . . 8 |- ( ( card ` x ) = x <-> ( x e. On /\ A. y e. x y ~< x ) )
5	2, 3, 4	3bitri 271	. . . . . . 7 |- ( x e. A <-> ( x e. On /\ A. y e. x y ~< x ) )
6	5	simplbi 460	. . . . . 6 |- ( x e. A -> x e. On )
7	6	ssriv 3508	. . . . 5 |- A C_ On
8		ssonuni 6601	. . . . 5 |- ( A e. _V -> ( A C_ On -> U. A e. On ) )
9	7, 8	mpi 17	. . . 4 |- ( A e. _V -> U. A e. On )
10		domrefg 7550	. . . . 5 |- ( U. A e. On -> U. A ~<_ U. A )
11	9, 10	syl 16	. . . 4 |- ( A e. _V -> U. A ~<_ U. A )
12		elharval 7988	. . . 4 |- ( U. A e. (har ` U. A ) <-> ( U. A e. On /\ U. A ~<_ U. A ) )
13	9, 11, 12	sylanbrc 664	. . 3 |- ( A e. _V -> U. A e. (har ` U. A ) )
14	7	sseli 3500	. . . . . . . 8 |- ( z e. A -> z e. On )
15		domrefg 7550	. . . . . . . . . 10 |- ( z e. On -> z ~<_ z )
16	15	ancli 551	. . . . . . . . 9 |- ( z e. On -> ( z e. On /\ z ~<_ z ) )
17		elharval 7988	. . . . . . . . 9 |- ( z e. (har ` z ) <-> ( z e. On /\ z ~<_ z ) )
18	16, 17	sylibr 212	. . . . . . . 8 |- ( z e. On -> z e. (har ` z ) )
19	14, 18	syl 16	. . . . . . 7 |- ( z e. A -> z e. (har ` z ) )
20		harcard 8358	. . . . . . . 8 |- ( card ` (har ` z ) ) = (har ` z )
21		fvex 5875	. . . . . . . . 9 |- (har ` z ) e. _V
22		fveq2 5865	. . . . . . . . . 10 |- ( x = (har ` z ) -> ( card ` x ) = ( card ` (har ` z ) ) )
23		id 22	. . . . . . . . . 10 |- ( x = (har ` z ) -> x = (har ` z ) )
24	22, 23	eqeq12d 2489	. . . . . . . . 9 |- ( x = (har ` z ) -> ( ( card ` x ) = x <-> ( card ` (har ` z ) ) = (har ` z ) ) )
25	21, 24, 1	elab2 3253	. . . . . . . 8 |- ( (har ` z ) e. A <-> ( card ` (har ` z ) ) = (har ` z ) )
26	20, 25	mpbir 209	. . . . . . 7 |- (har ` z ) e. A
27		eleq2 2540	. . . . . . . . 9 |- ( w = (har ` z ) -> ( z e. w <-> z e. (har ` z ) ) )
28		eleq1 2539	. . . . . . . . 9 |- ( w = (har ` z ) -> ( w e. A <-> (har ` z ) e. A ) )
29	27, 28	anbi12d 710	. . . . . . . 8 |- ( w = (har ` z ) -> ( ( z e. w /\ w e. A ) <-> ( z e. (har ` z ) /\ (har ` z ) e. A ) ) )
30	21, 29	spcev 3205	. . . . . . 7 |- ( ( z e. (har ` z ) /\ (har ` z ) e. A ) -> E. w ( z e. w /\ w e. A ) )
31	19, 26, 30	sylancl 662	. . . . . 6 |- ( z e. A -> E. w ( z e. w /\ w e. A ) )
32		eluni 4248	. . . . . 6 |- ( z e. U. A <-> E. w ( z e. w /\ w e. A ) )
33	31, 32	sylibr 212	. . . . 5 |- ( z e. A -> z e. U. A )
34	33	ssriv 3508	. . . 4 |- A C_ U. A
35		harcard 8358	. . . . 5 |- ( card ` (har ` U. A ) ) = (har ` U. A )
36		fvex 5875	. . . . . 6 |- (har ` U. A ) e. _V
37		fveq2 5865	. . . . . . 7 |- ( x = (har ` U. A ) -> ( card ` x ) = ( card ` (har ` U. A ) ) )
38		id 22	. . . . . . 7 |- ( x = (har ` U. A ) -> x = (har ` U. A ) )
39	37, 38	eqeq12d 2489	. . . . . 6 |- ( x = (har ` U. A ) -> ( ( card ` x ) = x <-> ( card ` (har ` U. A ) ) = (har ` U. A ) ) )
40	36, 39, 1	elab2 3253	. . . . 5 |- ( (har ` U. A ) e. A <-> ( card ` (har ` U. A ) ) = (har ` U. A ) )
41	35, 40	mpbir 209	. . . 4 |- (har ` U. A ) e. A
42	34, 41	sselii 3501	. . 3 |- (har ` U. A ) e. U. A
43	13, 42	jctir 538	. 2 |- ( A e. _V -> ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
44		eloni 4888	. . 3 |- ( U. A e. On -> Ord U. A )
45		ordn2lp 4898	. . 3 |- ( Ord U. A -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
46	9, 44, 45	3syl 20	. 2 |- ( A e. _V -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
47	43, 46	pm2.65i 173	1 |- -. A e. _V

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113   C_ wss 3476   U.cuni 4245   class class class wbr 4447   Ord word 4877   Oncon0 4878   ` cfv 5587   ~<_ cdom 7514   ~< csdm 7515  harchar 7981   cardccrd 8315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-recs 7042  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-oi 7934  df-har 7983  df-card 8319

This theorem is referenced by:  cardprc  8360