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Theorem tsk2 8477
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 8476 . 2  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  1o  e.  T )
2 df-2o 6567 . . 3  |-  2o  =  suc  1o
3 1on 6573 . . . 4  |-  1o  e.  On
4 tsksuc 8474 . . . 4  |-  ( ( T  e.  Tarski  /\  1o  e.  On  /\  1o  e.  T )  ->  suc  1o  e.  T )
53, 4mp3an2 1265 . . 3  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  suc  1o  e.  T )
62, 5syl5eqel 2442 . 2  |-  ( ( T  e.  Tarski  /\  1o  e.  T )  ->  2o  e.  T )
71, 6syldan 456 1  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  2o  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710    =/= wne 2521   (/)c0 3531   Oncon0 4474   suc csuc 4476   1oc1o 6559   2oc2o 6560   Tarskictsk 8460
This theorem is referenced by:  2domtsk  8478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-suc 4480  df-1o 6566  df-2o 6567  df-tsk 8461
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