Theorem tsken 10740

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 10736	. . . 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 270	. . 3 ⊢ (𝑇 ∈ Tarski → (∀𝑥 ∈ 𝑇 (𝒫 𝑥 ⊆ 𝑇 ∧ ∃𝑦 ∈ 𝑇 𝒫 𝑥 ⊆ 𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇)))
3	2	simprd 500	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))
4		elpw2g 5305	. . 3 ⊢ (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇 ↔ 𝐴 ⊆ 𝑇))
5	4	biimpar 482	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → 𝐴 ∈ 𝒫 𝑇)
6		breq1 5113	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑇 ↔ 𝐴 ≈ 𝑇))
7		eleq1 2851	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	orbi12d 931	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ↔ (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)))
9	8	rspccva 3581	. 2 ⊢ ((∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
10	3, 5, 9	syl2an2r 697	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3906  𝒫 cpw 4563   class class class wbr 5110   ≈ cen 8941  Tarskictsk 10734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-tsk 10735

This theorem is referenced by:  tskssel  10743  inttsk  10760  r1tskina  10768  tskuni  10769