Theorem disjx0 5063

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 5062	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5040	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

Syntax hints:   ⊆ wss 3939  ∅c0 4294  {csn 4570  Disj wdisj 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rmo 3149  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295  df-sn 4571  df-disj 5035

This theorem is referenced by: (None)