Theorem tsksdom 9146

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 7683	. . 3 |- ( A e. T -> A ~< ~P A )
2	1	adantl 466	. 2 |- ( ( T e. Tarski /\ A e. T ) -> A ~< ~P A )
3		simpl 457	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> T e. Tarski )
4		tskpwss 9142	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
5		ssdomg 7573	. . 3 |- ( T e. Tarski -> ( ~P A C_ T -> ~P A ~<_ T ) )
6	3, 4, 5	sylc 60	. 2 |- ( ( T e. Tarski /\ A e. T ) -> ~P A ~<_ T )
7		sdomdomtr 7662	. 2 |- ( ( A ~< ~P A /\ ~P A ~<_ T ) -> A ~< T )
8	2, 6, 7	syl2anc 661	1 |- ( ( T e. Tarski /\ A e. T ) -> A ~< T )

Syntax hints:   -> wi 4   /\ wa 369   e. wcel 1767   C_ wss 3481   ~Pcpw 4016   class class class wbr 4453   ~<_ cdom 7526   ~< csdm 7527   Tarskictsk 9138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-tsk 9139

This theorem is referenced by:  2domtsk  9156  r1tskina  9172  tskuni  9173  tskurn  9179  inaprc  9226