Theorem tsksdom 9180

Description: An element of a Tarski class is strictly dominated by the class. JFM CLASSES2 th. 1 (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.)

Assertion

Ref	Expression
tsksdom	|- ( ( T e. Tarski /\ A e. T ) -> A ~< T )

Proof of Theorem tsksdom

Step	Hyp	Ref	Expression
1		canth2g 7732	. . 3 |- ( A e. T -> A ~< ~P A )
2	1	adantl 467	. 2 |- ( ( T e. Tarski /\ A e. T ) -> A ~< ~P A )
3		simpl 458	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> T e. Tarski )
4		tskpwss 9176	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
5		ssdomg 7622	. . 3 |- ( T e. Tarski -> ( ~P A C_ T -> ~P A ~<_ T ) )
6	3, 4, 5	sylc 62	. 2 |- ( ( T e. Tarski /\ A e. T ) -> ~P A ~<_ T )
7		sdomdomtr 7711	. 2 |- ( ( A ~< ~P A /\ ~P A ~<_ T ) -> A ~< T )
8	2, 6, 7	syl2anc 665	1 |- ( ( T e. Tarski /\ A e. T ) -> A ~< T )

Syntax hints:   -> wi 4   /\ wa 370   e. wcel 1870   C_ wss 3442   ~Pcpw 3985   class class class wbr 4426   ~<_ cdom 7575   ~< csdm 7576   Tarskictsk 9172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-tsk 9173

This theorem is referenced by:  2domtsk  9190  r1tskina  9206  tskuni  9207  tskurn  9213  inaprc  9260