Theorem cardprclem 10060

Description: Lemma for cardprc 10061. (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 2852	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3		abid 2742	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4		iscard 10056	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
5	2, 3, 4	3bitri 300	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
6	5	simplbi 502	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
7	6	ssriv 3912	. . . . 5 ⊢ 𝐴 ⊆ On
8		ssonuni 7794	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
9	7, 8	mpi 21	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ On)
10		domrefg 9014	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ≼ ∪ 𝐴)
11	9, 10	syl 18	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ≼ ∪ 𝐴)
12		elharval 9555	. . . 4 ⊢ (∪ 𝐴 ∈ (har‘∪ 𝐴) ↔ (∪ 𝐴 ∈ On ∧ ∪ 𝐴 ≼ ∪ 𝐴))
13	9, 11, 12	sylanbrc 595	. . 3 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ (har‘∪ 𝐴))
14	7	sseli 3904	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ On)
15		domrefg 9014	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → 𝑧 ≼ 𝑧)
16	15	ancli 558	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
17		elharval 9555	. . . . . . . . 9 ⊢ (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
18	16, 17	sylibr 237	. . . . . . . 8 ⊢ (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
19	14, 18	syl 18	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (har‘𝑧))
20		harcard 10059	. . . . . . . 8 ⊢ (card‘(har‘𝑧)) = (har‘𝑧)
21		fvex 6898	. . . . . . . . 9 ⊢ (har‘𝑧) ∈ V
22		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23		id 23	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
24	22, 23	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
25	21, 24, 1	elab2 3635	. . . . . . . 8 ⊢ ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
26	20, 25	mpbir 234	. . . . . . 7 ⊢ (har‘𝑧) ∈ 𝐴
27		eleq2 2849	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (har‘𝑧)))
28		eleq1 2848	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑤 ∈ 𝐴 ↔ (har‘𝑧) ∈ 𝐴))
29	27, 28	anbi12d 644	. . . . . . . 8 ⊢ (𝑤 = (har‘𝑧) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
30	21, 29	spcev 3560	. . . . . . 7 ⊢ ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
31	19, 26, 30	sylancl 598	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
32		eluni 4869	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
33	31, 32	sylibr 237	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴)
34	33	ssriv 3912	. . . 4 ⊢ 𝐴 ⊆ ∪ 𝐴
35		harcard 10059	. . . . 5 ⊢ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)
36		fvex 6898	. . . . . 6 ⊢ (har‘∪ 𝐴) ∈ V
37		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → (card‘𝑥) = (card‘(har‘∪ 𝐴)))
38		id 23	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → 𝑥 = (har‘∪ 𝐴))
39	37, 38	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = (har‘∪ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)))
40	36, 39, 1	elab2 3635	. . . . 5 ⊢ ((har‘∪ 𝐴) ∈ 𝐴 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴))
41	35, 40	mpbir 234	. . . 4 ⊢ (har‘∪ 𝐴) ∈ 𝐴
42	34, 41	sselii 3905	. . 3 ⊢ (har‘∪ 𝐴) ∈ ∪ 𝐴
43	13, 42	jctir 530	. 2 ⊢ (𝐴 ∈ V → (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
44		eloni 6372	. . 3 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
45		ordn2lp 6382	. . 3 ⊢ (Ord ∪ 𝐴 → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
46	9, 44, 45	3syl 19	. 2 ⊢ (𝐴 ∈ V → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
47	43, 46	pm2.65i 196	1 ⊢ ¬ 𝐴 ∈ V

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ∪ cuni 4866   class class class wbr 5102  Ord word 6361  Oncon0 6362  ‘cfv 6538   ≼ cdom 8971   ≺ csdm 8972  harchar 9550  cardccrd 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-oi 9504  df-har 9551  df-card 10020

This theorem is used by:  cardprc  10061