Theorem disjx0 4368

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 3730	. 2 |- (/) C_ { (/) }
2		disjxsn 4367	. 2 |- Disj x e. { (/) } B
3		disjss1 4350	. 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 3371   (/)c0 3698   {csn 3935  Disj wdisj 4344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rmo 2744  df-v 3014  df-dif 3374  df-in 3378  df-ss 3385  df-nul 3699  df-sn 3936  df-disj 4345

This theorem is referenced by: (None)