Theorem disjx0 4432

Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjx0	|- Disj x e. (/) B

Proof of Theorem disjx0

Step	Hyp	Ref	Expression
1		0ss 3800	. 2 |- (/) C_ { (/) }
2		disjxsn 4431	. 2 |- Disj x e. { (/) } B
3		disjss1 4413	. 2 |- ( (/) C_ { (/) } -> (Disj x e. { (/) } B -> Disj x e. (/) B ) )
4	1, 2, 3	mp2 9	1 |- Disj x e. (/) B

Syntax hints:   C_ wss 3461   (/)c0 3770   {csn 4014  Disj wdisj 4407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rmo 2801  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3771  df-sn 4015  df-disj 4408

This theorem is referenced by: (None)