MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsksn Structured version   Visualization version   GIF version

Theorem tsksn 10800
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 10793 . 2 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
2 snsspw 4844 . . 3 {𝐴} ⊆ 𝒫 𝐴
3 tskss 10798 . . 3 ((𝑇 ∈ Tarski ∧ 𝒫 𝐴𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇)
42, 3mp3an3 1452 . 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 395  wcel 2108  wss 3951  𝒫 cpw 4600  {csn 4626  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-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pow 5365
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-nf 1784  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-tsk 10789
This theorem is referenced by:  tsk1  10804  tskop  10811
  Copyright terms: Public domain W3C validator