Theorem cardprclem 8137

Description: Lemma for cardprc 8138. (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 2497	. . . . . . . 8 |- ( x e. A <-> x e. { x | ( card ` x ) = x } )
3		abid 2421	. . . . . . . 8 |- ( x e. { x | ( card ` x ) = x } <-> ( card ` x ) = x )
4		iscard 8133	. . . . . . . 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 457	. . . . . 6 |- ( x e. A -> x e. On )
7	6	ssriv 3348	. . . . 5 |- A C_ On
8		ssonuni 6387	. . . . 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 7332	. . . . 5 |- ( U. A e. On -> U. A ~<_ U. A )
11	9, 10	syl 16	. . . 4 |- ( A e. _V -> U. A ~<_ U. A )
12		elharval 7766	. . . 4 |- ( U. A e. (har ` U. A ) <-> ( U. A e. On /\ U. A ~<_ U. A ) )
13	9, 11, 12	sylanbrc 657	. . 3 |- ( A e. _V -> U. A e. (har ` U. A ) )
14	7	sseli 3340	. . . . . . . 8 |- ( z e. A -> z e. On )
15		domrefg 7332	. . . . . . . . . 10 |- ( z e. On -> z ~<_ z )
16	15	ancli 546	. . . . . . . . 9 |- ( z e. On -> ( z e. On /\ z ~<_ z ) )
17		elharval 7766	. . . . . . . . 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 8136	. . . . . . . 8 |- ( card ` (har ` z ) ) = (har ` z )
21		fvex 5689	. . . . . . . . 9 |- (har ` z ) e. _V
22		fveq2 5679	. . . . . . . . . 10 |- ( x = (har ` z ) -> ( card ` x ) = ( card ` (har ` z ) ) )
23		id 22	. . . . . . . . . 10 |- ( x = (har ` z ) -> x = (har ` z ) )
24	22, 23	eqeq12d 2447	. . . . . . . . 9 |- ( x = (har ` z ) -> ( ( card ` x ) = x <-> ( card ` (har ` z ) ) = (har ` z ) ) )
25	21, 24, 1	elab2 3098	. . . . . . . 8 |- ( (har ` z ) e. A <-> ( card ` (har ` z ) ) = (har ` z ) )
26	20, 25	mpbir 209	. . . . . . 7 |- (har ` z ) e. A
27		eleq2 2494	. . . . . . . . 9 |- ( w = (har ` z ) -> ( z e. w <-> z e. (har ` z ) ) )
28		eleq1 2493	. . . . . . . . 9 |- ( w = (har ` z ) -> ( w e. A <-> (har ` z ) e. A ) )
29	27, 28	anbi12d 703	. . . . . . . 8 |- ( w = (har ` z ) -> ( ( z e. w /\ w e. A ) <-> ( z e. (har ` z ) /\ (har ` z ) e. A ) ) )
30	21, 29	spcev 3053	. . . . . . 7 |- ( ( z e. (har ` z ) /\ (har ` z ) e. A ) -> E. w ( z e. w /\ w e. A ) )
31	19, 26, 30	sylancl 655	. . . . . 6 |- ( z e. A -> E. w ( z e. w /\ w e. A ) )
32		eluni 4082	. . . . . 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 3348	. . . 4 |- A C_ U. A
35		harcard 8136	. . . . 5 |- ( card ` (har ` U. A ) ) = (har ` U. A )
36		fvex 5689	. . . . . 6 |- (har ` U. A ) e. _V
37		fveq2 5679	. . . . . . 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 2447	. . . . . 6 |- ( x = (har ` U. A ) -> ( ( card ` x ) = x <-> ( card ` (har ` U. A ) ) = (har ` U. A ) ) )
40	36, 39, 1	elab2 3098	. . . . 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 3341	. . 3 |- (har ` U. A ) e. U. A
43	13, 42	jctir 535	. 2 |- ( A e. _V -> ( U. A e. (har ` U. A ) /\ (har ` U. A ) e. U. A ) )
44		eloni 4716	. . 3 |- ( U. A e. On -> Ord U. A )
45		ordn2lp 4726	. . 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 1362   E.wex 1589   e. wcel 1755   {cab 2419   A.wral 2705   _Vcvv 2962   C_ wss 3316   U.cuni 4079   class class class wbr 4280   Ord word 4705   Oncon0 4706   ` cfv 5406   ~<_ cdom 7296   ~< csdm 7297  harchar 7759   cardccrd 8093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-recs 6818  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-oi 7712  df-har 7761  df-card 8097

This theorem is referenced by:  cardprc  8138