MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disj1 Structured version   Visualization version   GIF version

Theorem disj1 4403
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
disj1 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj1
StepHypRef Expression
1 disj 4401 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 df-ral 3147 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
31, 2bitri 276 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1528   = wceq 1530  wcel 2107  wral 3142  cin 3938  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-ral 3147  df-dif 3942  df-in 3946  df-nul 4295
This theorem is referenced by:  reldisj  4404  disj3  4405  undif4  4418  disjsn  4645  funun  6396  zfregs2  9167  dfac5lem4  9544  isf32lem9  9775  fzodisj  13064  fzodisjsn  13068  inpr0  30208  bnj1280  32178  ecin0  35476  zfregs2VD  41042
  Copyright terms: Public domain W3C validator