Theorem cardprclem 9872

Description: Lemma for cardprc 9873. (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 ∈ 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 2823	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3		abid 2713	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4		iscard 9868	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
5	2, 3, 4	3bitri 297	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
6	5	simplbi 497	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
7	6	ssriv 3933	. . . . 5 ⊢ 𝐴 ⊆ On
8		ssonuni 7713	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
9	7, 8	mpi 20	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ On)
10		domrefg 8909	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ≼ ∪ 𝐴)
11	9, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ≼ ∪ 𝐴)
12		elharval 9447	. . . 4 ⊢ (∪ 𝐴 ∈ (har‘∪ 𝐴) ↔ (∪ 𝐴 ∈ On ∧ ∪ 𝐴 ≼ ∪ 𝐴))
13	9, 11, 12	sylanbrc 583	. . 3 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ (har‘∪ 𝐴))
14	7	sseli 3925	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ On)
15		domrefg 8909	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → 𝑧 ≼ 𝑧)
16	15	ancli 548	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
17		elharval 9447	. . . . . . . . 9 ⊢ (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
18	16, 17	sylibr 234	. . . . . . . 8 ⊢ (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
19	14, 18	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (har‘𝑧))
20		harcard 9871	. . . . . . . 8 ⊢ (card‘(har‘𝑧)) = (har‘𝑧)
21		fvex 6835	. . . . . . . . 9 ⊢ (har‘𝑧) ∈ V
22		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
24	22, 23	eqeq12d 2747	. . . . . . . . 9 ⊢ (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
25	21, 24, 1	elab2 3633	. . . . . . . 8 ⊢ ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
26	20, 25	mpbir 231	. . . . . . 7 ⊢ (har‘𝑧) ∈ 𝐴
27		eleq2 2820	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (har‘𝑧)))
28		eleq1 2819	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑤 ∈ 𝐴 ↔ (har‘𝑧) ∈ 𝐴))
29	27, 28	anbi12d 632	. . . . . . . 8 ⊢ (𝑤 = (har‘𝑧) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
30	21, 29	spcev 3556	. . . . . . 7 ⊢ ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
31	19, 26, 30	sylancl 586	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
32		eluni 4859	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
33	31, 32	sylibr 234	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴)
34	33	ssriv 3933	. . . 4 ⊢ 𝐴 ⊆ ∪ 𝐴
35		harcard 9871	. . . . 5 ⊢ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)
36		fvex 6835	. . . . . 6 ⊢ (har‘∪ 𝐴) ∈ V
37		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → (card‘𝑥) = (card‘(har‘∪ 𝐴)))
38		id 22	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → 𝑥 = (har‘∪ 𝐴))
39	37, 38	eqeq12d 2747	. . . . . 6 ⊢ (𝑥 = (har‘∪ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)))
40	36, 39, 1	elab2 3633	. . . . 5 ⊢ ((har‘∪ 𝐴) ∈ 𝐴 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴))
41	35, 40	mpbir 231	. . . 4 ⊢ (har‘∪ 𝐴) ∈ 𝐴
42	34, 41	sselii 3926	. . 3 ⊢ (har‘∪ 𝐴) ∈ ∪ 𝐴
43	13, 42	jctir 520	. 2 ⊢ (𝐴 ∈ V → (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
44		eloni 6316	. . 3 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
45		ordn2lp 6326	. . 3 ⊢ (Ord ∪ 𝐴 → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
46	9, 44, 45	3syl 18	. 2 ⊢ (𝐴 ∈ V → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
47	43, 46	pm2.65i 194	1 ⊢ ¬ 𝐴 ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  ∪ cuni 4856   class class class wbr 5089  Ord word 6305  Oncon0 6306  ‘cfv 6481   ≼ cdom 8867   ≺ csdm 8868  harchar 9442  cardccrd 9828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-oi 9396  df-har 9443  df-card 9832

This theorem is referenced by:  cardprc  9873