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Theorem tskssel 10797
Description: A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskssel ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)

Proof of Theorem tskssel
StepHypRef Expression
1 sdomnen 9021 . . 3 (𝐴𝑇 → ¬ 𝐴𝑇)
213ad2ant3 1136 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → ¬ 𝐴𝑇)
3 tsken 10794 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
433adant3 1133 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (𝐴𝑇𝐴𝑇))
54ord 865 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (¬ 𝐴𝑇𝐴𝑇))
62, 5mpd 15 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 848  w3a 1087  wcel 2108  wss 3951   class class class wbr 5143  cen 8982  csdm 8984  Tarskictsk 10788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-sdom 8988  df-tsk 10789
This theorem is referenced by:  tskpr  10810  tskwe2  10813  tskord  10820  tskcard  10821  tskurn  10829
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