Theorem eltsk2g 9828

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 9827	. 2 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
2		nfra1 3088	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇
3		pweq 4320	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
4	3	sseq1d 3794	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ⊆ 𝑇 ↔ 𝒫 𝑤 ⊆ 𝑇))
5	4	rspccva 3461	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
6	5	adantlr 706	. . . . . . . . 9 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
7		vpwex 5015	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
8	7	elpw 4323	. . . . . . . . . 10 ⊢ (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤)
9		ssel 3757	. . . . . . . . . 10 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧 ∈ 𝑇))
10	8, 9	syl5bir 234	. . . . . . . . 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 3178	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) → (∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
13	2, 12	ralimdaa 3105	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 → (∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
14	13	imdistani 564	. . . . 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 3211	. . . . 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 3211	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ↔ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
17	14, 15, 16	3imtr4i 283	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
18		ssid 3785	. . . . . . 7 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
19		sseq2 3789	. . . . . . . 8 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
20	19	rspcev 3462	. . . . . . 7 ⊢ ((𝒫 𝑧 ∈ 𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
21	18, 20	mpan2 682	. . . . . 6 ⊢ (𝒫 𝑧 ∈ 𝑇 → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
22	21	anim2i 610	. . . . 5 ⊢ ((𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
23	22	ralimi 3099	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
24	17, 23	impbii 200	. . 3 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
25	24	anbi1i 617	. 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 278	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 197   ∧ wa 384   ∨ wo 873   ∈ wcel 2155  ∀wral 3055  ∃wrex 3056   ⊆ wss 3734  𝒫 cpw 4317   class class class wbr 4811   ≈ cen 8159  Tarskictsk 9825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-pow 5003

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-tsk 9826

This theorem is referenced by:  tskpw  9830  0tsk  9832  inttsk  9851  inatsk  9855