Theorem eltskg 10679

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 3970	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝒫 𝑧 ⊆ 𝑦 ↔ 𝒫 𝑧 ⊆ 𝑇))
2		rexeq 3292	. . . . 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 3308	. . 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 4573	. . . 4 ⊢ (𝑦 = 𝑇 → 𝒫 𝑦 = 𝒫 𝑇)
6		breq2 5106	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑧 ≈ 𝑦 ↔ 𝑧 ≈ 𝑇))
7		eleq2 2817	. . . . 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 3316	. . 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 10678	. 2 ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))}
12	10, 11	elab2g 3644	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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  𝒫 cpw 4559   class class class wbr 5102   ≈ cen 8892  Tarskictsk 10677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-tsk 10678

This theorem is referenced by:  eltsk2g  10680  tskpwss  10681  tsken  10683  grothtsk  10764