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Theorem tsk0 10780
Description: A nonempty Tarski class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tsk0 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)

Proof of Theorem tsk0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 n0 4342 . . 3 (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥𝑇)
2 0ss 4392 . . . . . 6 ∅ ⊆ 𝑥
3 tskss 10775 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑥𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
42, 3mp3an3 1447 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∅ ∈ 𝑇)
54expcom 413 . . . 4 (𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
65exlimiv 1926 . . 3 (∃𝑥 𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
71, 6sylbi 216 . 2 (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
87impcom 407 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1774  wcel 2099  wne 2936  wss 3945  c0 4318  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-ext 2699  ax-sep 5293
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  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-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-tsk 10766
This theorem is referenced by:  tsk1  10781  tskr1om  10784
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