Theorem tsken 10832

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 10828	. . . 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 501	. 2 ⊢ (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇))
4		elpw2g 5295	. . 3 ⊢ (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇 ↔ 𝐴 ⊆ 𝑇))
5	4	biimpar 483	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → 𝐴 ∈ 𝒫 𝑇)
6		breq1 5106	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑇 ↔ 𝐴 ≈ 𝑇))
7		eleq1 2849	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑇 ↔ 𝐴 ∈ 𝑇))
8	6, 7	orbi12d 932	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ↔ (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)))
9	8	rspccva 3576	. 2 ⊢ ((∀𝑥 ∈ 𝒫 𝑇(𝑥 ≈ 𝑇 ∨ 𝑥 ∈ 𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))
10	3, 5, 9	syl2an2r 698	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103   ≈ cen 8963  Tarskictsk 10826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-tsk 10827

This theorem is used by:  tskssel  10835  inttsk  10852  r1tskina  10860  tskuni  10861