Theorem tsken 10713

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 10709	. . . 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 267	. . 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 5290	. . 3 ⊢ (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇 ↔ 𝐴 ⊆ 𝑇))
5	4	biimpar 477	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → 𝐴 ∈ 𝒫 𝑇)
6		breq1 5112	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑇 ↔ 𝐴 ≈ 𝑇))
7		eleq1 2817	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	orbi12d 918	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ↔ (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)))
9	8	rspccva 3590	. 2 ⊢ ((∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
10	3, 5, 9	syl2an2r 685	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3916  𝒫 cpw 4565   class class class wbr 5109   ≈ cen 8917  Tarskictsk 10707

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 2702  ax-sep 5253

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 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-tsk 10708

This theorem is referenced by:  tskssel  10716  inttsk  10733  r1tskina  10741  tskuni  10742