Theorem cardprclem 9976

Description: Lemma for cardprc 9977. (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 2711	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4		iscard 9972	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
5	2, 3, 4	3bitri 296	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
6	5	simplbi 496	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
7	6	ssriv 3985	. . . . 5 ⊢ 𝐴 ⊆ On
8		ssonuni 7769	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
9	7, 8	mpi 20	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ On)
10		domrefg 8985	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ≼ ∪ 𝐴)
11	9, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ≼ ∪ 𝐴)
12		elharval 9558	. . . 4 ⊢ (∪ 𝐴 ∈ (har‘∪ 𝐴) ↔ (∪ 𝐴 ∈ On ∧ ∪ 𝐴 ≼ ∪ 𝐴))
13	9, 11, 12	sylanbrc 581	. . 3 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ (har‘∪ 𝐴))
14	7	sseli 3977	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ On)
15		domrefg 8985	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → 𝑧 ≼ 𝑧)
16	15	ancli 547	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
17		elharval 9558	. . . . . . . . 9 ⊢ (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
18	16, 17	sylibr 233	. . . . . . . 8 ⊢ (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
19	14, 18	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (har‘𝑧))
20		harcard 9975	. . . . . . . 8 ⊢ (card‘(har‘𝑧)) = (har‘𝑧)
21		fvex 6903	. . . . . . . . 9 ⊢ (har‘𝑧) ∈ V
22		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
24	22, 23	eqeq12d 2746	. . . . . . . . 9 ⊢ (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
25	21, 24, 1	elab2 3671	. . . . . . . 8 ⊢ ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
26	20, 25	mpbir 230	. . . . . . 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 629	. . . . . . . 8 ⊢ (𝑤 = (har‘𝑧) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
30	21, 29	spcev 3595	. . . . . . 7 ⊢ ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
31	19, 26, 30	sylancl 584	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
32		eluni 4910	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
33	31, 32	sylibr 233	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴)
34	33	ssriv 3985	. . . 4 ⊢ 𝐴 ⊆ ∪ 𝐴
35		harcard 9975	. . . . 5 ⊢ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)
36		fvex 6903	. . . . . 6 ⊢ (har‘∪ 𝐴) ∈ V
37		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → (card‘𝑥) = (card‘(har‘∪ 𝐴)))
38		id 22	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → 𝑥 = (har‘∪ 𝐴))
39	37, 38	eqeq12d 2746	. . . . . 6 ⊢ (𝑥 = (har‘∪ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)))
40	36, 39, 1	elab2 3671	. . . . 5 ⊢ ((har‘∪ 𝐴) ∈ 𝐴 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴))
41	35, 40	mpbir 230	. . . 4 ⊢ (har‘∪ 𝐴) ∈ 𝐴
42	34, 41	sselii 3978	. . 3 ⊢ (har‘∪ 𝐴) ∈ ∪ 𝐴
43	13, 42	jctir 519	. 2 ⊢ (𝐴 ∈ V → (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
44		eloni 6373	. . 3 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
45		ordn2lp 6383	. . 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 193	1 ⊢ ¬ 𝐴 ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707  ∀wral 3059  Vcvv 3472   ⊆ wss 3947  ∪ cuni 4907   class class class wbr 5147  Ord word 6362  Oncon0 6363  ‘cfv 6542   ≼ cdom 8939   ≺ csdm 8940  harchar 9553  cardccrd 9932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-har 9554  df-card 9936

This theorem is referenced by:  cardprc  9977