Theorem cardprclem 8141

Description: Lemma for cardprc 8142. (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 2502	. . . . . . . 8 |- ( x e. A <-> x e. { x | ( card ` x ) = x } )
3		abid 2426	. . . . . . . 8 |- ( x e. { x | ( card ` x ) = x } <-> ( card ` x ) = x )
4		iscard 8137	. . . . . . . 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 3355	. . . . 5 |- A C_ On
8		ssonuni 6393	. . . . 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 7336	. . . . 5 |- ( U. A e. On -> U. A ~<_ U. A )
11	9, 10	syl 16	. . . 4 |- ( A e. _V -> U. A ~<_ U. A )
12		elharval 7770	. . . 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 3347	. . . . . . . 8 |- ( z e. A -> z e. On )
15		domrefg 7336	. . . . . . . . . 10 |- ( z e. On -> z ~<_ z )
16	15	ancli 551	. . . . . . . . 9 |- ( z e. On -> ( z e. On /\ z ~<_ z ) )
17		elharval 7770	. . . . . . . . 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 8140	. . . . . . . 8 |- ( card ` (har ` z ) ) = (har ` z )
21		fvex 5696	. . . . . . . . 9 |- (har ` z ) e. _V
22		fveq2 5686	. . . . . . . . . 10 |- ( x = (har ` z ) -> ( card ` x ) = ( card ` (har ` z ) ) )
23		id 22	. . . . . . . . . 10 |- ( x = (har ` z ) -> x = (har ` z ) )
24	22, 23	eqeq12d 2452	. . . . . . . . 9 |- ( x = (har ` z ) -> ( ( card ` x ) = x <-> ( card ` (har ` z ) ) = (har ` z ) ) )
25	21, 24, 1	elab2 3104	. . . . . . . 8 |- ( (har ` z ) e. A <-> ( card ` (har ` z ) ) = (har ` z ) )
26	20, 25	mpbir 209	. . . . . . 7 |- (har ` z ) e. A
27		eleq2 2499	. . . . . . . . 9 |- ( w = (har ` z ) -> ( z e. w <-> z e. (har ` z ) ) )
28		eleq1 2498	. . . . . . . . 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 3059	. . . . . . 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 4089	. . . . . 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 3355	. . . 4 |- A C_ U. A
35		harcard 8140	. . . . 5 |- ( card ` (har ` U. A ) ) = (har ` U. A )
36		fvex 5696	. . . . . 6 |- (har ` U. A ) e. _V
37		fveq2 5686	. . . . . . 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 2452	. . . . . 6 |- ( x = (har ` U. A ) -> ( ( card ` x ) = x <-> ( card ` (har ` U. A ) ) = (har ` U. A ) ) )
40	36, 39, 1	elab2 3104	. . . . 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 3348	. . 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 4724	. . 3 |- ( U. A e. On -> Ord U. A )
45		ordn2lp 4734	. . 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 1369   E.wex 1586   e. wcel 1756   {cab 2424   A.wral 2710   _Vcvv 2967   C_ wss 3323   U.cuni 4086   class class class wbr 4287   Ord word 4713   Oncon0 4714   ` cfv 5413   ~<_ cdom 7300   ~< csdm 7301  harchar 7763   cardccrd 8097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-recs 6824  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-oi 7716  df-har 7765  df-card 8101

This theorem is referenced by:  cardprc  8142