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Theorem tskssel 9781
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 8138 . . 3 (𝐴𝑇 → ¬ 𝐴𝑇)
213ad2ant3 1129 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → ¬ 𝐴𝑇)
3 tsken 9778 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
433adant3 1126 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (𝐴𝑇𝐴𝑇))
54ord 853 . 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 836  w3a 1071  wcel 2145  wss 3723   class class class wbr 4786  cen 8106  csdm 8108  Tarskictsk 9772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-sdom 8112  df-tsk 9773
This theorem is referenced by:  tskpr  9794  tskwe2  9797  tskord  9804  tskcard  9805  tskurn  9813
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