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Theorem tsksdom 10167
Description: An element of a Tarski class is strictly dominated by the class. JFM CLASSES2 th. 1. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsksdom ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)

Proof of Theorem tsksdom
StepHypRef Expression
1 canth2g 8659 . 2 (𝐴𝑇𝐴 ≺ 𝒫 𝐴)
2 simpl 486 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝑇 ∈ Tarski)
3 tskpwss 10163 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
4 ssdomg 8542 . . 3 (𝑇 ∈ Tarski → (𝒫 𝐴𝑇 → 𝒫 𝐴𝑇))
52, 3, 4sylc 65 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
6 sdomdomtr 8638 . 2 ((𝐴 ≺ 𝒫 𝐴 ∧ 𝒫 𝐴𝑇) → 𝐴𝑇)
71, 5, 6syl2an2 685 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  wss 3908  𝒫 cpw 4511   class class class wbr 5042  cdom 8494  csdm 8495  Tarskictsk 10159
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-tsk 10160
This theorem is referenced by:  2domtsk  10177  r1tskina  10193  tskuni  10194  tskurn  10200  inaprc  10247
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