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Theorem tsk2 8596
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 8595 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )
2 df-2o 6684 . . 3  |-  2o  =  suc  1o
3 1on 6690 . . . 4  |-  1o  e.  On
4 tsksuc 8593 . . . 4  |-  ( ( T  e.  Tarski  /\  1o  e.  On  /\  1o  e.  T )  ->  suc  1o  e.  T )
53, 4mp3an2 1267 . . 3  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  suc  1o  e.  T )
62, 5syl5eqel 2488 . 2  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  2o  e.  T )
71, 6syldan 457 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    =/= wne 2567   (/)c0 3588   Oncon0 4541   suc csuc 4543   1oc1o 6676   2oc2o 6677   Tarskictsk 8579
This theorem is referenced by:  2domtsk  8597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-1o 6683  df-2o 6684  df-tsk 8580
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