Theorem disjx0 4434

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 3813	. 2 |- (/) C_ { (/) }
2		disjxsn 4433	. 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 3461   (/)c0 3783   {csn 4016  Disj wdisj 4410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rmo 2812  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-disj 4411

This theorem is referenced by: (None)