Theorem tsken 10675

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 10671	. . . 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 268	. . 3 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simprd 496	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))
4		elpw2g 5268	. . 3 ⊢ (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇 ↔ 𝐴 ⊆ 𝑇))
5	4	biimpar 478	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → 𝐴 ∈ 𝒫 𝑇)
6		breq1 5082	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑇 ↔ 𝐴 ≈ 𝑇))
7		eleq1 2828	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	orbi12d 924	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ↔ (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)))
9	8	rspccva 3566	. 2 ⊢ ((∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
10	3, 5, 9	syl2an2r 691	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ⊆ wss 3890  𝒫 cpw 4536   class class class wbr 5079   ≈ cen 8887  Tarskictsk 10669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-tsk 10670

This theorem is referenced by:  tskssel  10678  inttsk  10695  r1tskina  10703  tskuni  10704