Theorem cardprclem 8431

Description: Lemma for cardprc 8432. (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 2541	. . . . . . . 8 |- ( x e. A <-> x e. { x | ( card ` x ) = x } )
3		abid 2459	. . . . . . . 8 |- ( x e. { x | ( card ` x ) = x } <-> ( card ` x ) = x )
4		iscard 8427	. . . . . . . 8 |- ( ( card ` x ) = x <-> ( x e. On /\ A. y e. x y ~< x ) )
5	2, 3, 4	3bitri 279	. . . . . . 7 |- ( x e. A <-> ( x e. On /\ A. y e. x y ~< x ) )
6	5	simplbi 467	. . . . . 6 |- ( x e. A -> x e. On )
7	6	ssriv 3422	. . . . 5 |- A C_ On
8		ssonuni 6632	. . . . 5 |- ( A e. _V -> ( A C_ On -> U. A e. On ) )
9	7, 8	mpi 20	. . . 4 |- ( A e. _V -> U. A e. On )
10		domrefg 7622	. . . . 5 |- ( U. A e. On -> U. A ~<_ U. A )
11	9, 10	syl 17	. . . 4 |- ( A e. _V -> U. A ~<_ U. A )
12		elharval 8096	. . . 4 |- ( U. A e. (har ` U. A ) <-> ( U. A e. On /\ U. A ~<_ U. A ) )
13	9, 11, 12	sylanbrc 677	. . 3 |- ( A e. _V -> U. A e. (har ` U. A ) )
14	7	sseli 3414	. . . . . . . 8 |- ( z e. A -> z e. On )
15		domrefg 7622	. . . . . . . . . 10 |- ( z e. On -> z ~<_ z )
16	15	ancli 560	. . . . . . . . 9 |- ( z e. On -> ( z e. On /\ z ~<_ z ) )
17		elharval 8096	. . . . . . . . 9 |- ( z e. (har ` z ) <-> ( z e. On /\ z ~<_ z ) )
18	16, 17	sylibr 217	. . . . . . . 8 |- ( z e. On -> z e. (har ` z ) )
19	14, 18	syl 17	. . . . . . 7 |- ( z e. A -> z e. (har ` z ) )
20		harcard 8430	. . . . . . . 8 |- ( card ` (har ` z ) ) = (har ` z )
21		fvex 5889	. . . . . . . . 9 |- (har ` z ) e. _V
22		fveq2 5879	. . . . . . . . . 10 |- ( x = (har ` z ) -> ( card ` x ) = ( card ` (har ` z ) ) )
23		id 22	. . . . . . . . . 10 |- ( x = (har ` z ) -> x = (har ` z ) )
24	22, 23	eqeq12d 2486	. . . . . . . . 9 |- ( x = (har ` z ) -> ( ( card ` x ) = x <-> ( card ` (har ` z ) ) = (har ` z ) ) )
25	21, 24, 1	elab2 3176	. . . . . . . 8 |- ( (har ` z ) e. A <-> ( card ` (har ` z ) ) = (har ` z ) )
26	20, 25	mpbir 214	. . . . . . 7 |- (har ` z ) e. A
27		eleq2 2538	. . . . . . . . 9 |- ( w = (har ` z ) -> ( z e. w <-> z e. (har ` z ) ) )
28		eleq1 2537	. . . . . . . . 9 |- ( w = (har ` z ) -> ( w e. A <-> (har ` z ) e. A ) )
29	27, 28	anbi12d 725	. . . . . . . 8 |- ( w = (har ` z ) -> ( ( z e. w /\ w e. A ) <-> ( z e. (har ` z ) /\ (har ` z ) e. A ) ) )
30	21, 29	spcev 3127	. . . . . . 7 |- ( ( z e. (har ` z ) /\ (har ` z ) e. A ) -> E. w ( z e. w /\ w e. A ) )
31	19, 26, 30	sylancl 675	. . . . . 6 |- ( z e. A -> E. w ( z e. w /\ w e. A ) )
32		eluni 4193	. . . . . 6 |- ( z e. U. A <-> E. w ( z e. w /\ w e. A ) )
33	31, 32	sylibr 217	. . . . 5 |- ( z e. A -> z e. U. A )
34	33	ssriv 3422	. . . 4 |- A C_ U. A
35		harcard 8430	. . . . 5 |- ( card ` (har ` U. A ) ) = (har ` U. A )
36		fvex 5889	. . . . . 6 |- (har ` U. A ) e. _V
37		fveq2 5879	. . . . . . 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 2486	. . . . . 6 |- ( x = (har ` U. A ) -> ( ( card ` x ) = x <-> ( card ` (har ` U. A ) ) = (har ` U. A ) ) )
40	36, 39, 1	elab2 3176	. . . . 5 |- ( (har ` U. A ) e. A <-> ( card ` (har ` U. A ) ) = (har ` U. A ) )
41	35, 40	mpbir 214	. . . 4 |- (har ` U. A ) e. A
42	34, 41	sselii 3415	. . 3 |- (har ` U. A ) e. U. A
43	13, 42	jctir 547	. 2 |- ( A e. _V -> ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
44		eloni 5440	. . 3 |- ( U. A e. On -> Ord U. A )
45		ordn2lp 5450	. . 3 |- ( Ord U. A -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
46	9, 44, 45	3syl 18	. 2 |- ( A e. _V -> -. ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
47	43, 46	pm2.65i 178	1 |- -. A e. _V

Syntax hints:   -. wn 3   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   {cab 2457   A.wral 2756   _Vcvv 3031   C_ wss 3390   U.cuni 4190   class class class wbr 4395   Ord word 5429   Oncon0 5430   ` cfv 5589   ~<_ cdom 7585   ~< csdm 7586  harchar 8089   cardccrd 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-wrecs 7046  df-recs 7108  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-oi 8043  df-har 8091  df-card 8391

This theorem is referenced by:  cardprc  8432