Theorem cardprclem 7822

Description: Lemma for cardprc 7823. (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 2468	. . . . . . . 8 |- ( x e. A <-> x e. { x | ( card ` x ) = x } )
3		abid 2392	. . . . . . . 8 |- ( x e. { x | ( card ` x ) = x } <-> ( card ` x ) = x )
4		iscard 7818	. . . . . . . 8 |- ( ( card ` x ) = x <-> ( x e. On /\ A. y e. x y ~< x ) )
5	2, 3, 4	3bitri 263	. . . . . . 7 |- ( x e. A <-> ( x e. On /\ A. y e. x y ~< x ) )
6	5	simplbi 447	. . . . . 6 |- ( x e. A -> x e. On )
7	6	ssriv 3312	. . . . 5 |- A C_ On
8		ssonuni 4726	. . . . 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 7101	. . . . 5 |- ( U. A e. On -> U. A ~<_ U. A )
11	9, 10	syl 16	. . . 4 |- ( A e. _V -> U. A ~<_ U. A )
12		elharval 7487	. . . 4 |- ( U. A e. (har ` U. A ) <-> ( U. A e. On /\ U. A ~<_ U. A ) )
13	9, 11, 12	sylanbrc 646	. . 3 |- ( A e. _V -> U. A e. (har ` U. A ) )
14	7	sseli 3304	. . . . . . . 8 |- ( z e. A -> z e. On )
15		domrefg 7101	. . . . . . . . . 10 |- ( z e. On -> z ~<_ z )
16	15	ancli 535	. . . . . . . . 9 |- ( z e. On -> ( z e. On /\ z ~<_ z ) )
17		elharval 7487	. . . . . . . . 9 |- ( z e. (har ` z ) <-> ( z e. On /\ z ~<_ z ) )
18	16, 17	sylibr 204	. . . . . . . 8 |- ( z e. On -> z e. (har ` z ) )
19	14, 18	syl 16	. . . . . . 7 |- ( z e. A -> z e. (har ` z ) )
20		harcard 7821	. . . . . . . 8 |- ( card ` (har ` z ) ) = (har ` z )
21		fvex 5701	. . . . . . . . 9 |- (har ` z ) e. _V
22		fveq2 5687	. . . . . . . . . 10 |- ( x = (har ` z ) -> ( card ` x ) = ( card ` (har ` z ) ) )
23		id 20	. . . . . . . . . 10 |- ( x = (har ` z ) -> x = (har ` z ) )
24	22, 23	eqeq12d 2418	. . . . . . . . 9 |- ( x = (har ` z ) -> ( ( card ` x ) = x <-> ( card ` (har ` z ) ) = (har ` z ) ) )
25	21, 24, 1	elab2 3045	. . . . . . . 8 |- ( (har ` z ) e. A <-> ( card ` (har ` z ) ) = (har ` z ) )
26	20, 25	mpbir 201	. . . . . . 7 |- (har ` z ) e. A
27		eleq2 2465	. . . . . . . . 9 |- ( w = (har ` z ) -> ( z e. w <-> z e. (har ` z ) ) )
28		eleq1 2464	. . . . . . . . 9 |- ( w = (har ` z ) -> ( w e. A <-> (har ` z ) e. A ) )
29	27, 28	anbi12d 692	. . . . . . . 8 |- ( w = (har ` z ) -> ( ( z e. w /\ w e. A ) <-> ( z e. (har ` z ) /\ (har ` z ) e. A ) ) )
30	21, 29	spcev 3003	. . . . . . 7 |- ( ( z e. (har ` z ) /\ (har ` z ) e. A ) -> E. w ( z e. w /\ w e. A ) )
31	19, 26, 30	sylancl 644	. . . . . 6 |- ( z e. A -> E. w ( z e. w /\ w e. A ) )
32		eluni 3978	. . . . . 6 |- ( z e. U. A <-> E. w ( z e. w /\ w e. A ) )
33	31, 32	sylibr 204	. . . . 5 |- ( z e. A -> z e. U. A )
34	33	ssriv 3312	. . . 4 |- A C_ U. A
35		harcard 7821	. . . . 5 |- ( card ` (har ` U. A ) ) = (har ` U. A )
36		fvex 5701	. . . . . 6 |- (har ` U. A ) e. _V
37		fveq2 5687	. . . . . . 7 |- ( x = (har ` U. A ) -> ( card ` x ) = ( card ` (har ` U. A ) ) )
38		id 20	. . . . . . 7 |- ( x = (har ` U. A ) -> x = (har ` U. A ) )
39	37, 38	eqeq12d 2418	. . . . . 6 |- ( x = (har ` U. A ) -> ( ( card ` x ) = x <-> ( card ` (har ` U. A ) ) = (har ` U. A ) ) )
40	36, 39, 1	elab2 3045	. . . . 5 |- ( (har ` U. A ) e. A <-> ( card ` (har ` U. A ) ) = (har ` U. A ) )
41	35, 40	mpbir 201	. . . 4 |- (har ` U. A ) e. A
42	34, 41	sselii 3305	. . 3 |- (har ` U. A ) e. U. A
43	13, 42	jctir 525	. 2 |- ( A e. _V -> ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
44		eloni 4551	. . 3 |- ( U. A e. On -> Ord U. A )
45		ordn2lp 4561	. . 3 |- ( Ord U. A -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
46	9, 44, 45	3syl 19	. 2 |- ( A e. _V -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
47	43, 46	pm2.65i 167	1 |- -. A e. _V

Syntax hints:   -. wn 3   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   {cab 2390   A.wral 2666   _Vcvv 2916   C_ wss 3280   U.cuni 3975   class class class wbr 4172   Ord word 4540   Oncon0 4541   ` cfv 5413   ~<_ cdom 7066   ~< csdm 7067  harchar 7480   cardccrd 7778

This theorem is referenced by:  cardprc  7823

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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 6508  df-recs 6592  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-oi 7435  df-har 7482  df-card 7782