Theorem tsksn 9582

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 e. Tarski /\ A e. T ) -> { A } e. T )

Proof of Theorem tsksn

Step	Hyp	Ref	Expression
1		tskpw 9575	. 2 |- ( ( T e. Tarski /\ A e. T ) -> ~P A e. T )
2		snsspw 4375	. . 3 |- { A } C_ ~P A
3		tskss 9580	. . 3 |- ( ( T e. Tarski /\ ~P A e. T /\ { A } C_ ~P A ) -> { A } e. T )
4	2, 3	mp3an3 1413	. 2 |- ( ( T e. Tarski /\ ~P A e. T ) -> { A } e. T )
5	1, 4	syldan 487	1 |- ( ( T e. Tarski /\ A e. T ) -> { A } e. T )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   {csn 4177   Tarskictsk 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-tsk 9571

This theorem is referenced by:  tsk1  9586  tskop  9593