Theorem eltsk2g 10664

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 10663	. 2 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
2		nfra1 3260	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇
3		pweq 4568	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
4	3	sseq1d 3965	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ⊆ 𝑇 ↔ 𝒫 𝑤 ⊆ 𝑇))
5	4	rspccva 3575	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
6	5	adantlr 715	. . . . . . . . 9 ⊢ (((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ 𝑤 ∈ 𝑇) → 𝒫 𝑤 ⊆ 𝑇)
7		vpwex 5322	. . . . . . . . . . 11 ⊢ 𝒫 𝑧 ∈ V
8	7	elpw 4558	. . . . . . . . . 10 ⊢ (𝒫 𝑧 ∈ 𝒫 𝑤 ↔ 𝒫 𝑧 ⊆ 𝑤)
9		ssel 3927	. . . . . . . . . 10 ⊢ (𝒫 𝑤 ⊆ 𝑇 → (𝒫 𝑧 ∈ 𝒫 𝑤 → 𝒫 𝑧 ∈ 𝑇))
10	8, 9	biimtrrid 243	. . . . . . . . 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 3137	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ 𝑧 ∈ 𝑇) → (∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → 𝒫 𝑧 ∈ 𝑇))
13	2, 12	ralimdaa 3237	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 → (∀𝑧 ∈ 𝑇 ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤 → ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
14	13	imdistani 568	. . . . 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 3096	. . . . 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 3096	. . . . 5 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ↔ (∀𝑧 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑇 ∧ ∀𝑧 ∈ 𝑇 𝒫 𝑧 ∈ 𝑇))
17	14, 15, 16	3imtr4i 292	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
18		ssid 3956	. . . . . . 7 ⊢ 𝒫 𝑧 ⊆ 𝒫 𝑧
19		sseq2 3960	. . . . . . . 8 ⊢ (𝑤 = 𝒫 𝑧 → (𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝑧 ⊆ 𝒫 𝑧))
20	19	rspcev 3576	. . . . . . 7 ⊢ ((𝒫 𝑧 ∈ 𝑇 ∧ 𝒫 𝑧 ⊆ 𝒫 𝑧) → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
21	18, 20	mpan2 691	. . . . . 6 ⊢ (𝒫 𝑧 ∈ 𝑇 → ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)
22	21	anim2i 617	. . . . 5 ⊢ ((𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
23	22	ralimi 3073	. . . 4 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) → ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
24	17, 23	impbii 209	. . 3 ⊢ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇))
25	24	anbi1i 624	. 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 287	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 206   ∧ wa 395   ∨ wo 847   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098   ≈ cen 8882  Tarskictsk 10661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-pow 5310

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-tsk 10662

This theorem is referenced by:  tskpw  10666  0tsk  10668  inttsk  10687  inatsk  10691