Theorem eltskg 10862

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 3939	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑧 ⊆ 𝑇))
2		rexeq 3319	. . . . 5 ⊢ (𝑦 = 𝑇 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)))
4	3	raleqbi1dv 3333	. . 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 4574	. . . 4 ⊢ (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6		breq2 5110	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ≈ 𝑦 ↔ 𝑧 ≈ 𝑇))
7		eleq2 2853	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑇))
8	6, 7	orbi12d 932	. . . 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 644	. 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 10861	. 2 ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
12	10, 11	elab2g 3637	1 ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106   ≈ cen 8998  Tarskictsk 10860

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-tsk 10861

This theorem is used by:  eltsk2g  10863  tskpwss  10864  tsken  10866  grothtsk  10947