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Theorem tsken 9831
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 9827 . . . 4 (𝑇 ∈ Tarski → (𝑇 ∈ Tarski ↔ (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))))
21ibi 258 . . 3 (𝑇 ∈ Tarski → (∀𝑥𝑇 (𝒫 𝑥𝑇 ∧ ∃𝑦𝑇 𝒫 𝑥𝑦) ∧ ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇)))
32simprd 489 . 2 (𝑇 ∈ Tarski → ∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇))
4 elpw2g 4987 . . 3 (𝑇 ∈ Tarski → (𝐴 ∈ 𝒫 𝑇𝐴𝑇))
54biimpar 469 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴 ∈ 𝒫 𝑇)
6 breq1 4814 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
7 eleq1 2832 . . . 4 (𝑥 = 𝐴 → (𝑥𝑇𝐴𝑇))
86, 7orbi12d 942 . . 3 (𝑥 = 𝐴 → ((𝑥𝑇𝑥𝑇) ↔ (𝐴𝑇𝐴𝑇)))
98rspccva 3461 . 2 ((∀𝑥 ∈ 𝒫 𝑇(𝑥𝑇𝑥𝑇) ∧ 𝐴 ∈ 𝒫 𝑇) → (𝐴𝑇𝐴𝑇))
103, 5, 9syl2an2r 675 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3734  𝒫 cpw 4317   class class class wbr 4811  cen 8159  Tarskictsk 9825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-tsk 9826
This theorem is referenced by:  tskssel  9834  inttsk  9851  r1tskina  9859  tskuni  9860
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