Theorem 0disj 5070

Description: Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
0disj	⊢ Disj 𝑥 ∈ 𝐴 ∅

Proof of Theorem 0disj

Step	Hyp	Ref	Expression
1		0ss 4335	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 3077	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 5069	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 5046	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 3065   ⊆ wss 3891  ∅c0 4261  {csn 4566  Disj wdisj 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-mo 2541  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rmo 3073  df-v 3432  df-dif 3894  df-in 3898  df-ss 3908  df-nul 4262  df-sn 4567  df-disj 5044

This theorem is referenced by: (None)