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Theorem tskssel 9458
 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 7870 . . 3 (𝐴𝑇 → ¬ 𝐴𝑇)
213ad2ant3 1077 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → ¬ 𝐴𝑇)
3 tsken 9455 . . . 4 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
433adant3 1074 . . 3 ((𝑇 ∈ Tarski ∧ 𝐴𝑇𝐴𝑇) → (𝐴𝑇𝐴𝑇))
54ord 391 . 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 382   ∧ w3a 1031   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583   ≈ cen 7838   ≺ csdm 7840  Tarskictsk 9449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-sdom 7844  df-tsk 9450 This theorem is referenced by:  tskpr  9471  tskwe2  9474  tskord  9481  tskcard  9482  tskurn  9490
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