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 ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)

Proof of Theorem tsksn

Step	Hyp	Ref	Expression
1		tskpw 10793	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇)
2		snsspw 4844	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
3		tskss 10798	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇 ∧ {𝐴} ⊆ 𝒫 𝐴) → {𝐴} ∈ 𝑇)
4	2, 3	mp3an3 1452	. 2 ⊢ ((𝑇 ∈ Tarski ∧ 𝒫 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)
5	1, 4	syldan 591	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇)

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