Theorem disjx0 5106

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 4357	. 2 ⊢ ∅ ⊆ {∅}
2		disjxsn 5105	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5084	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

This proof depends on syntax axioms:   ⊆ wss 3906  ∅c0 4286  {csn 4591  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rmo 3371  df-dif 3909  df-ss 3923  df-nul 4287  df-sn 4592  df-disj 5079

This theorem is used by: (None)