Theorem eltsk2g 10755

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 10754	. 2 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
2		nfra1 3291	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇
3		pweq 4578	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
4	3	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ⊆ 𝑇 ↔ 𝒫 𝑤 ⊆ 𝑇))
5	4	rspccva 3582	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
6	5	adantlr 728	. . . . . . . . 9 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
7		vpwex 5350	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
8	7	elpw 4568	. . . . . . . . . 10 ⊢ (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤)
9		ssel 3932	. . . . . . . . . 10 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧 ∈ 𝑇))
10	8, 9	biimtrrid 246	. . . . . . . . 9 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
11	6, 10	syl 18	. . . . . . . 8 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
12	11	rexlimdva 3168	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) → (∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
13	2, 12	ralimdaa 3268	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 → (∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
14	13	imdistani 579	. . . . 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 3127	. . . . 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 3127	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ↔ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
17	14, 15, 16	3imtr4i 295	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
18		ssid 3960	. . . . . . 7 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
19		sseq2 3964	. . . . . . . 8 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
20	19	rspcev 3583	. . . . . . 7 ⊢ ((𝒫 𝑧 ∈ 𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
21	18, 20	mpan2 704	. . . . . 6 ⊢ (𝒫 𝑧 ∈ 𝑇 → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
22	21	anim2i 629	. . . . 5 ⊢ ((𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
23	22	ralimi 3104	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
24	17, 23	impbii 212	. . 3 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
25	24	anbi1i 636	. 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	bitrdi 290	1 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  𝒫 cpw 4564   class class class wbr 5111   ≈ cen 8946  Tarskictsk 10752

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-tsk 10753

This theorem is used by:  tskpw  10757  0tsk  10759  inttsk  10778  inatsk  10782