Theorem 0disj 4645

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 3972	. . 3 ⊢ ∅ ⊆ {𝑥}
2	1	rgenw 2924	. 2 ⊢ ∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥}
3		sndisj 4644	. 2 ⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
4		disjss2 4623	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∅ ⊆ {𝑥} → (Disj 𝑥 ∈ 𝐴 {𝑥} → Disj 𝑥 ∈ 𝐴 ∅))
5	2, 3, 4	mp2 9	1 ⊢ Disj 𝑥 ∈ 𝐴 ∅

Syntax hints:  ∀wral 2912   ⊆ wss 3574  ∅c0 3915  {csn 4177  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-disj 4621

This theorem is referenced by: (None)