Theorem disjx0 5137

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 4399	. 2 ⊢ ∅ ⊆ {∅}
2		disjxsn 5136	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5115	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

Syntax hints:   ⊆ wss 3950  ∅c0 4332  {csn 4625  Disj wdisj 5109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-rmo 3379  df-dif 3953  df-ss 3967  df-nul 4333  df-sn 4626  df-disj 5110

This theorem is referenced by: (None)