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Theorem tsk0 10183
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 4293 . . 3 (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥𝑇)
2 0ss 4333 . . . . . 6 ∅ ⊆ 𝑥
3 tskss 10178 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑥𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
42, 3mp3an3 1447 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∅ ∈ 𝑇)
54expcom 417 . . . 4 (𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
65exlimiv 1932 . . 3 (∃𝑥 𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
71, 6sylbi 220 . 2 (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
87impcom 411 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1781  wcel 2115  wne 3014  wss 3919  c0 4276  Tarskictsk 10168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-tsk 10169
This theorem is referenced by:  tsk1  10184  tskr1om  10187
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