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

Theorem disj 3969
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
disj ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj
StepHypRef Expression
1 df-in 3547 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
21eqeq1i 2615 . . 3 ((𝐴𝐵) = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅)
3 abeq1 2720 . . 3 ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅ ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
4 imnan 437 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
5 noel 3878 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 361 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
74, 6bitr2i 264 . . . 4 (((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
87albii 1737 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
92, 3, 83bitri 285 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 df-ral 2901 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
119, 10bitr4i 266 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wcel 1977  {cab 2596  wral 2896  cin 3539  c0 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-in 3547  df-nul 3875
This theorem is referenced by:  disjr  3970  disj1  3971  disjne  3974  otiunsndisj  4905  onxpdisj  5764  f0rn0  6003  onint  6887  zfreg  8383  zfregOLD  8385  kmlem4  8858  fin23lem30  9047  fin23lem31  9048  isf32lem3  9060  fpwwe2  9344  renfdisj  9977  fvinim0ffz  12449  s3iunsndisj  13555  metdsge  22460  spthispth  26103  2spotdisj  26588  2spotiundisj  26589  2spotmdisj  26595  subfacp1lem1  30415  dfpo2  30898  dvmptfprodlem  38834  stoweidlem26  38919  stoweidlem59  38952  iundjiunlem  39352  otiunsndisjX  40317  2wspdisj  41165  2wspiundisj  41166  2wspmdisj  41501
  Copyright terms: Public domain W3C validator