MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsk1 Structured version   Unicode version

Theorem tsk1 8640
Description: One is an element of a non-empty Tarski's class. (Contributed by FL, 22-Feb-2011.)
Assertion
Ref Expression
tsk1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )

Proof of Theorem tsk1
StepHypRef Expression
1 df1o2 6737 . 2  |-  1o  =  { (/) }
2 tsk0 8639 . . 3  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  (/)  e.  T
)
3 tsksn 8636 . . 3  |-  ( ( T  e.  Tarski  /\  (/)  e.  T
)  ->  { (/) }  e.  T )
42, 3syldan 458 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  { (/) }  e.  T )
51, 4syl5eqel 2521 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726    =/= wne 2600   (/)c0 3629   {csn 3815   1oc1o 6718   Tarskictsk 8624
This theorem is referenced by:  tsk2  8641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-pow 4378
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-suc 4588  df-1o 6725  df-tsk 8625
  Copyright terms: Public domain W3C validator