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Theorem tsk2 10757
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 10756 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o𝑇)
2 df-2o 8463 . . 3 2o = suc 1o
3 1on 8474 . . . 4 1o ∈ On
4 tsksuc 10754 . . . 4 ((𝑇 ∈ Tarski ∧ 1o ∈ On ∧ 1o𝑇) → suc 1o𝑇)
53, 4mp3an2 1445 . . 3 ((𝑇 ∈ Tarski ∧ 1o𝑇) → suc 1o𝑇)
62, 5eqeltrid 2829 . 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 2098  wne 2932  c0 4315  Oncon0 6355  suc csuc 6357  1oc1o 8455  2oc2o 8456  Tarskictsk 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359  df-suc 6361  df-1o 8462  df-2o 8463  df-tsk 10741
This theorem is referenced by:  2domtsk  10758
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