ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  int0el GIF version

Theorem int0el 3673
Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
int0el (∅ ∈ 𝐴 𝐴 = ∅)

Proof of Theorem int0el
StepHypRef Expression
1 intss1 3658 . 2 (∅ ∈ 𝐴 𝐴 ⊆ ∅)
2 0ss 3283 . . 3 ∅ ⊆ 𝐴
32a1i 9 . 2 (∅ ∈ 𝐴 → ∅ ⊆ 𝐴)
41, 3eqssd 2990 1 (∅ ∈ 𝐴 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wss 2945  c0 3252   cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253  df-int 3644
This theorem is referenced by:  intv  3951  inton  4158
  Copyright terms: Public domain W3C validator