Theorem tsk0 10519

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.

Step	Hyp	Ref	Expression
1		n0 4280	. . 3 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
2		0ss 4330	. . . . . 6 ⊢ ∅ ⊆ 𝑥
3		tskss 10514	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
4	2, 3	mp3an3 1449	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → ∅ ∈ 𝑇)
5	4	expcom 414	. . . 4 ⊢ (𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
6	5	exlimiv 1933	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
7	1, 6	sylbi 216	. 2 ⊢ (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
8	7	impcom 408	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943   ⊆ wss 3887  ∅c0 4256  Tarskictsk 10504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-tsk 10505

This theorem is referenced by:  tsk1  10520  tskr1om  10523