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Theorem tsk2 9625
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 9624 . 2 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)
2 df-2o 7606 . . 3 2𝑜 = suc 1𝑜
3 1on 7612 . . . 4 1𝑜 ∈ On
4 tsksuc 9622 . . . 4 ((𝑇 ∈ Tarski ∧ 1𝑜 ∈ On ∧ 1𝑜𝑇) → suc 1𝑜𝑇)
53, 4mp3an2 1452 . . 3 ((𝑇 ∈ Tarski ∧ 1𝑜𝑇) → suc 1𝑜𝑇)
62, 5syl5eqel 2734 . 2 ((𝑇 ∈ Tarski ∧ 1𝑜𝑇) → 2𝑜𝑇)
71, 6syldan 486 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wne 2823  c0 3948  Oncon0 5761  suc csuc 5763  1𝑜c1o 7598  2𝑜c2o 7599  Tarskictsk 9608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606  df-tsk 9609
This theorem is referenced by:  2domtsk  9626
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