MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsk0 Structured version   Visualization version   GIF version

Theorem tsk0 9529
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 3907 . . 3 (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥𝑇)
2 0ss 3944 . . . . . 6 ∅ ⊆ 𝑥
3 tskss 9524 . . . . . 6 ((𝑇 ∈ Tarski ∧ 𝑥𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
42, 3mp3an3 1410 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝑥𝑇) → ∅ ∈ 𝑇)
54expcom 451 . . . 4 (𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
65exlimiv 1855 . . 3 (∃𝑥 𝑥𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
71, 6sylbi 207 . 2 (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
87impcom 446 1 ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1701  wcel 1987  wne 2790  wss 3555  c0 3891  Tarskictsk 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-tsk 9515
This theorem is referenced by:  tsk1  9530  tskr1om  9533
  Copyright terms: Public domain W3C validator