Theorem eltskg 10710

Description: Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)

Assertion

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

Distinct variable group:   𝑤,𝑇,𝑧

Allowed substitution hints:   𝑉(𝑧,𝑤)

Proof of Theorem eltskg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3964	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑧 ⊆ 𝑇))
2		rexeq 3318	. . . . 5 ⊢ (𝑦 = 𝑇 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
3	1, 2	anbi12d 641	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)))
4	3	raleqbi1dv 3332	. . 3 ⊢ (𝑦 = 𝑇 → (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ ∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)))
5		pweq 4571	. . . 4 ⊢ (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6		breq2 5106	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ≈ 𝑦 ↔ 𝑧 ≈ 𝑇))
7		eleq2 2853	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑇))
8	6, 7	orbi12d 929	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))
9	5, 8	raleqbidv 3338	. . 3 ⊢ (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))
10	4, 9	anbi12d 641	. 2 ⊢ (𝑦 = 𝑇 → ((∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))
11		df-tsk 10709	. 2 ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
12	10, 11	elab2g 3641	1 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906  𝒫 cpw 4557   class class class wbr 5102   ≈ cen 8926  Tarskictsk 10708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-tsk 10709

This theorem is referenced by:  eltsk2g  10711  tskpwss  10712  tsken  10714  grothtsk  10795