Theorem 0disj 5100

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 4363	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 3048	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 5099	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 5077	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 3044   ⊆ wss 3914  ∅c0 4296  {csn 4589  Disj wdisj 5074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rmo 3354  df-v 3449  df-dif 3917  df-ss 3931  df-nul 4297  df-sn 4590  df-disj 5075

This theorem is referenced by: (None)