Theorem 0disj 5106

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 4364	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 3089	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 5105	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 5083	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 3085   ⊆ wss 3913  ∅c0 4294  {csn 4594  Disj wdisj 5080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rmo 3376  df-dif 3916  df-ss 3930  df-nul 4295  df-sn 4595  df-disj 5081

This theorem is referenced by: (None)