Theorem 0disj 5093

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 4354	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 3080	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 5092	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 5070	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 3076   ⊆ wss 3904  ∅c0 4285  {csn 4582  Disj wdisj 5067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rmo 3367  df-dif 3907  df-ss 3921  df-nul 4286  df-sn 4583  df-disj 5068

This theorem is referenced by: (None)