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Theorem tsk2 8671
Description: Two is an element of a non-empty Tarski's class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsk2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  e.  T )

Proof of Theorem tsk2
StepHypRef Expression
1 tsk1 8670 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )
2 df-2o 6754 . . 3  |-  2o  =  suc  1o
3 1on 6760 . . . 4  |-  1o  e.  On
4 tsksuc 8668 . . . 4  |-  ( ( T  e.  Tarski  /\  1o  e.  On  /\  1o  e.  T )  ->  suc  1o  e.  T )
53, 4mp3an2 1268 . . 3  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  suc  1o  e.  T )
62, 5syl5eqel 2526 . 2  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  2o  e.  T )
71, 6syldan 458 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727    =/= wne 2605   (/)c0 3613   Oncon0 4610   suc csuc 4612   1oc1o 6746   2oc2o 6747   Tarskictsk 8654
This theorem is referenced by:  2domtsk  8672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-suc 4616  df-1o 6753  df-2o 6754  df-tsk 8655
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