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Theorem tsken 10167
 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 10163 . . . 4 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 270 . . 3 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simprd 499 . 2 (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))
4 elpw2g 5211 . . 3 (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇𝐴𝑇))
54biimpar 481 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴 ∈ 𝒫 𝑇)
6 breq1 5033 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
7 eleq1 2877 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7orbi12d 916 . . 3 (𝑥 = 𝐴 → ((𝑥𝑇𝑥𝑇) ↔ (𝐴𝑇𝐴𝑇)))
98rspccva 3570 . 2 ((∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴𝑇𝐴𝑇))
103, 5, 9syl2an2r 684 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  𝒫 cpw 4497   class class class wbr 5030   ≈ cen 8491  Tarskictsk 10161 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-tsk 10162 This theorem is referenced by:  tskssel  10170  inttsk  10187  r1tskina  10195  tskuni  10196
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