Theorem disjx0 4014

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 3473	. 2 |- (/) C_ { (/) }
2		disjxsn 4013	. 2 |- Disj x e. { (/) } B
3		disjss1 3998	. 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 3141   (/)c0 3434   {csn 3604  Disj wdisj 3992

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rmo 2473  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-disj 3993

This theorem is referenced by: (None)