Theorem 0disj 5024

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 4292	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 3082	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 5023	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 5000	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 3070   ⊆ wss 3858  ∅c0 4225  {csn 4522  Disj wdisj 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2557  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rmo 3078  df-v 3411  df-dif 3861  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-disj 4998

This theorem is referenced by: (None)