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Theorem tsken 9788
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 ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))

Proof of Theorem tsken
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltskg 9784 . . . 4 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 256 . . 3 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simprd 482 . 2 (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))
4 elpw2g 4976 . . 3 (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇𝐴𝑇))
54biimpar 503 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴 ∈ 𝒫 𝑇)
6 breq1 4807 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
7 eleq1 2827 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7orbi12d 748 . . 3 (𝑥 = 𝐴 → ((𝑥𝑇𝑥𝑇) ↔ (𝐴𝑇𝐴𝑇)))
98rspccva 3448 . 2 ((∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴𝑇𝐴𝑇))
103, 5, 9syl2an2r 911 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715  𝒫 cpw 4302   class class class wbr 4804  cen 8120  Tarskictsk 9782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-tsk 9783
This theorem is referenced by:  tskssel  9791  inttsk  9808  r1tskina  9816  tskuni  9817
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