Theorem tsksdom 9033

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 7574	. . 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 9029	. . 3 |- ( ( T e. Tarski /\ A e. T ) -> ~P A C_ T )
5		ssdomg 7464	. . 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 7553	. 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 1758   C_ wss 3435   ~Pcpw 3967   class class class wbr 4399   ~<_ cdom 7417   ~< csdm 7418   Tarskictsk 9025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-tsk 9026

This theorem is referenced by:  2domtsk  9043  r1tskina  9059  tskuni  9060  tskurn  9066  inaprc  9113