Theorem tsk0 10755

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 4339	. . 3 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
2		0ss 4389	. . . . . 6 ⊢ ∅ ⊆ 𝑥
3		tskss 10750	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
4	2, 3	mp3an3 1446	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → ∅ ∈ 𝑇)
5	4	expcom 413	. . . 4 ⊢ (𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
6	5	exlimiv 1925	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
7	1, 6	sylbi 216	. 2 ⊢ (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
8	7	impcom 407	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1773   ∈ wcel 2098   ≠ wne 2932   ⊆ wss 3941  ∅c0 4315  Tarskictsk 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-tsk 10741

This theorem is referenced by:  tsk1  10756  tskr1om  10759