Theorem disjx0 5068

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 4329	. 2 ⊢ ∅ ⊆ {∅}
2		disjxsn 5067	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5046	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

Syntax hints:   ⊆ wss 3883  ∅c0 4262  {csn 4556  Disj wdisj 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-rmo 3344  df-dif 3886  df-ss 3900  df-nul 4263  df-sn 4557  df-disj 5041

This theorem is referenced by: (None)