Theorem disjx0 5104

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 5103	. 2 ⊢ Disj 𝑥 ∈ {∅}𝐵
3		disjss1 5082	. 2 ⊢ (∅ ⊆ {∅} → (Disj 𝑥 ∈ {∅}𝐵 → Disj 𝑥 ∈ ∅ 𝐵))
4	1, 2, 3	mp2 9	1 ⊢ Disj 𝑥 ∈ ∅ 𝐵

This proof depends on syntax axioms:   ⊆ wss 3905  ∅c0 4286  {csn 4589  Disj wdisj 5076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-rmo 3369  df-dif 3908  df-ss 3922  df-nul 4287  df-sn 4590  df-disj 5077

This theorem is used by: (None)