Theorem disjx0 5133

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 4389	. 2 ⊢ ∅ ⊆ {∅}
2		disjxsn 5132	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5110	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

Syntax hints:   ⊆ wss 3941  ∅c0 4315  {csn 4621  Disj wdisj 5104

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-rmo 3368  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-sn 4622  df-disj 5105

This theorem is referenced by: (None)