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Theorem tsksn 10170
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 10163 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
2 snsspw 4767 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 tskss 10168 . . 3 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇)
42, 3mp3an3 1441 . 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 2105  wss 3933  𝒫 cpw 4535  {csn 4557  Tarskictsk 10158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-pow 5257
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-tsk 10159
This theorem is referenced by:  tsk1  10174  tskop  10181
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