Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjx0 GIF version

Theorem disjx0 3754
 Description: An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjx0 Disj x B

Proof of Theorem disjx0
StepHypRef Expression
1 0ss 3249 . 2 ∅ ⊆ {∅}
2 disjxsn 3753 . 2 Disj x {∅}B
3 disjss1 3742 . 2 (∅ ⊆ {∅} → (Disj x {∅}BDisj x B))
41, 2, 3mp2 16 1 Disj x B
 Colors of variables: wff set class Syntax hints:   ⊆ wss 2911  ∅c0 3218  {csn 3367  Disj wdisj 3736 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rmo 2308  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-disj 3737 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator