Theorem eltskg 10641

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 3956	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑧 ⊆ 𝑇))
2		rexeq 3288	. . . . 5 ⊢ (𝑦 = 𝑇 → (∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤))
3	1, 2	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ↔ (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤)))
4	3	raleqbi1dv 3304	. . 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 4561	. . . 4 ⊢ (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6		breq2 5093	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ≈ 𝑦 ↔ 𝑧 ≈ 𝑇))
7		eleq2 2820	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑇))
8	6, 7	orbi12d 918	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ (𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))
9	5, 8	raleqbidv 3312	. . 3 ⊢ (𝑦 = 𝑇 → (∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))
10	4, 9	anbi12d 632	. 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 10640	. 2 ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
12	10, 11	elab2g 3631	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 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4547   class class class wbr 5089   ≈ cen 8866  Tarskictsk 10639

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 2113  ax-9 2121  ax-ext 2703

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-tsk 10640

This theorem is referenced by:  eltsk2g  10642  tskpwss  10643  tsken  10645  grothtsk  10726