Theorem tskpwss 9151

Description: First axiom of a Tarski class. The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tskpwss

Proof of Theorem tskpwss

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltskg 9149	. . . . 5
2	1	ibi 241	. . . 4
3	2	simpld 459	. . 3
4		simpl 457	. . . 4
5	4	ralimi 2850	. . 3
6	3, 5	syl 16	. 2
7		pweq 4015	. . . 4
8	7	sseq1d 3530	. . 3
9	8	rspccva 3209	. 2
10	6, 9	sylan 471	1

Syntax hints:  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  ~Pcpw 4012   class class class wbr 4452  cen 7533  ctsk 9147

This theorem is referenced by:  tsksdom  9155  tskss  9157  tsktrss  9160  inttsk  9173  tskcard  9180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-tsk 9148