Theorem disjx0 5101

Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjx0	⊢ Disj 𝑥 ∈ ∅ 𝐵

Proof of Theorem disjx0

Step	Hyp	Ref	Expression
1		0ss 4353	. 2 ⊢ ∅ ⊆ {∅}
2		disjxsn 5100	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5079	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

This proof depends on syntax axioms:   ⊆ wss 3900  ∅c0 4281  {csn 4588  Disj wdisj 5073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-rmo 3369  df-ss 3901  df-dif 4050  df-nul 4282  df-sn 4589  df-disj 5074

This theorem is used by: (None)