Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsken Unicode version

Theorem tsken 9153
 Description: Third axiom of a Tarski class. A subset of a Tarski class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsken

Proof of Theorem tsken
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpw2g 4615 . . 3
21biimpar 485 . 2
3 eltskg 9149 . . . . 5
43ibi 241 . . . 4
54simprd 463 . . 3
6 breq1 4455 . . . . 5
7 eleq1 2529 . . . . 5
86, 7orbi12d 709 . . . 4
98rspccva 3209 . . 3
105, 9sylan 471 . 2
112, 10syldan 470 1
 Colors of variables: wff setvar class 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:  tskssel  9156  inttsk  9173  r1tskina  9181  tskuni  9182 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  ax-sep 4573 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
 Copyright terms: Public domain W3C validator