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Theorem tsk2 10782
Description: Two is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsk2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o𝑇)

Proof of Theorem tsk2
StepHypRef Expression
1 tsk1 10781 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o𝑇)
2 df-2o 8481 . . 3 2o = suc 1o
3 1on 8492 . . . 4 1o ∈ On
4 tsksuc 10779 . . . 4 ((𝑇 ∈ Tarski ∧ 1o ∈ On ∧ 1o𝑇) → suc 1o𝑇)
53, 4mp3an2 1446 . . 3 ((𝑇 ∈ Tarski ∧ 1o𝑇) → suc 1o𝑇)
62, 5eqeltrid 2833 . 2 ((𝑇 ∈ Tarski ∧ 1o𝑇) → 2o𝑇)
71, 6syldan 590 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2099  wne 2936  c0 4318  Oncon0 6363  suc csuc 6365  1oc1o 8473  2oc2o 8474  Tarskictsk 10765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369  df-1o 8480  df-2o 8481  df-tsk 10766
This theorem is referenced by:  2domtsk  10783
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