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Theorem tsksn 10739
Description: A singleton of an element of a Tarski class belongs to the class. JFM CLASSES2 th. 2 (partly). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsksn ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → {𝐴} ∈ 𝑇)

Proof of Theorem tsksn
StepHypRef Expression
1 tskpw 10732 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
2 snsspw 4839 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 tskss 10737 . . 3 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇)
42, 3mp3an3 1450 . 2 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇) → {𝐴} ∈ 𝑇)
51, 4syldan 591 1 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → {𝐴} ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wss 3945  𝒫 cpw 4597  {csn 4623  Tarskictsk 10727
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-pow 5357
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-tsk 10728
This theorem is referenced by:  tsk1  10743  tskop  10750
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