Theorem tsk0 10718

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 4305	. . 3 ⊢ (𝑇 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑇)
2		0ss 4353	. . . . . 6 ⊢ ∅ ⊆ 𝑥
3		tskss 10713	. . . . . 6 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇 ∧ ∅ ⊆ 𝑥) → ∅ ∈ 𝑇)
4	2, 3	mp3an3 1470	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → ∅ ∈ 𝑇)
5	4	expcom 417	. . . 4 ⊢ (𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
6	5	exlimiv 1949	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝑇 → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
7	1, 6	sylbi 219	. 2 ⊢ (𝑇 ≠ ∅ → (𝑇 ∈ Tarski → ∅ ∈ 𝑇))
8	7	impcom 411	1 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956   ⊆ wss 3904  ∅c0 4285  Tarskictsk 10703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-tsk 10704

This theorem is referenced by:  tsk1  10719  tskr1om  10722