Theorem disjx0 4017

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 3476	. 2 |- (/) C_ { (/) }
2		disjxsn 4016	. 2 |- Disj x e. { (/) } B
3		disjss1 4001	. 2 |- ( (/) C_ { (/) } -> (Disj x e. { (/) } B -> Disj x e. (/) B ) )
4	1, 2, 3	mp2 16	1 |- Disj x e. (/) B

Syntax hints:   C_ wss 3144   (/)c0 3437   {csn 3607  Disj wdisj 3995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rmo 2476  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-disj 3996

This theorem is referenced by: (None)