Theorem tsksn 10447

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 ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)

Proof of Theorem tsksn

Step	Hyp	Ref	Expression
1		tskpw 10440	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
2		snsspw 4772	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
3		tskss 10445	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇)
4	2, 3	mp3an3 1448	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)
5	1, 4	syldan 590	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  Tarskictsk 10435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-pow 5283

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-tsk 10436

This theorem is referenced by:  tsk1  10451  tskop  10458