Theorem disjx0 4435

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 3807	. 2 |- (/) C_ { (/) }
2		disjxsn 4434	. 2 |- Disj x e. { (/) } B
3		disjss1 4416	. 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 3469   (/)c0 3778   {csn 4020  Disj wdisj 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rmo 2815  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483  df-nul 3779  df-sn 4021  df-disj 4411

This theorem is referenced by: (None)