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Theorem tsk2 9771
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 9770 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)
2 df-2o 7722 . . 3 2𝑜 = suc 1𝑜
3 1on 7728 . . . 4 1𝑜 ∈ On
4 tsksuc 9768 . . . 4 ((𝑇 ∈ Tarski ∧ 1𝑜 ∈ On ∧ 1𝑜𝑇) → suc 1𝑜𝑇)
53, 4mp3an2 1553 . . 3 ((𝑇 ∈ Tarski ∧ 1𝑜𝑇) → suc 1𝑜𝑇)
62, 5syl5eqel 2835 . 2 ((𝑇 ∈ Tarski ∧ 1𝑜𝑇) → 2𝑜𝑇)
71, 6syldan 488 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2131  wne 2924  c0 4050  Oncon0 5876  suc csuc 5878  1𝑜c1o 7714  2𝑜c2o 7715  Tarskictsk 9754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-ord 5879  df-on 5880  df-suc 5882  df-1o 7721  df-2o 7722  df-tsk 9755
This theorem is referenced by:  2domtsk  9772
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