Theorem eltsk2g 10172

Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
eltsk2g	⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))

Distinct variable group:   𝑧,𝑇

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem eltsk2g

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltskg 10171	. 2 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
2		nfra1 3219	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇
3		pweq 4554	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
4	3	sseq1d 3997	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ⊆ 𝑇 ↔ 𝒫 𝑤 ⊆ 𝑇))
5	4	rspccva 3621	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
6	5	adantlr 713	. . . . . . . . 9 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
7		vpwex 5277	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
8	7	elpw 4542	. . . . . . . . . 10 ⊢ (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤)
9		ssel 3960	. . . . . . . . . 10 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧 ∈ 𝑇))
10	8, 9	syl5bir 245	. . . . . . . . 9 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
11	6, 10	syl 17	. . . . . . . 8 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
12	11	rexlimdva 3284	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) → (∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
13	2, 12	ralimdaa 3217	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 → (∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
14	13	imdistani 571	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) → (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
15		r19.26 3170	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ↔ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
16		r19.26 3170	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ↔ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
17	14, 15, 16	3imtr4i 294	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
18		ssid 3988	. . . . . . 7 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
19		sseq2 3992	. . . . . . . 8 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
20	19	rspcev 3622	. . . . . . 7 ⊢ ((𝒫 𝑧 ∈ 𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
21	18, 20	mpan2 689	. . . . . 6 ⊢ (𝒫 𝑧 ∈ 𝑇 → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
22	21	anim2i 618	. . . . 5 ⊢ ((𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
23	22	ralimi 3160	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
24	17, 23	impbii 211	. . 3 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
25	24	anbi1i 625	. 2 ⊢ ((∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)) ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))
26	1, 25	syl6bb 289	1 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  𝒫 cpw 4538   class class class wbr 5065   ≈ cen 8505  Tarskictsk 10169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-pow 5265

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-tsk 10170

This theorem is referenced by:  tskpw  10174  0tsk  10176  inttsk  10195  inatsk  10199