Theorem tsken 10441

Description: Third axiom of a Tarski class. A subset of a Tarski class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tsken	⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

Proof of Theorem tsken

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltskg 10437	. . . 4 ⊢ (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))))
2	1	ibi 266	. . 3 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simprd 495	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))
4		elpw2g 5263	. . 3 ⊢ (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇 ↔ 𝐴 ⊆ 𝑇))
5	4	biimpar 477	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → 𝐴 ∈ 𝒫 𝑇)
6		breq1 5073	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑇 ↔ 𝐴 ≈ 𝑇))
7		eleq1 2826	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	orbi12d 915	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ↔ (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)))
9	8	rspccva 3551	. 2 ⊢ ((∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
10	3, 5, 9	syl2an2r 681	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070   ≈ cen 8688  Tarskictsk 10435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-tsk 10436

This theorem is referenced by:  tskssel  10444  inttsk  10461  r1tskina  10469  tskuni  10470