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

Theorem disj 3463
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 3127 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
21eqeq1i 2178 . . 3 ((𝐴𝐵) = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅)
3 abeq1 2280 . . 3 ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅ ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
4 imnan 685 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
5 noel 3418 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 694 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
74, 6bitr2i 184 . . . 4 (((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
87albii 1463 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
92, 3, 83bitri 205 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 df-ral 2453 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
119, 10bitr4i 186 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wcel 2141  {cab 2156  wral 2448  cin 3120  c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415
This theorem is referenced by:  disjr  3464  disj1  3465  disjne  3468  f0rn0  5392  renfdisj  7979  fvinim0ffz  10197  fxnn0nninf  10394  fprodsplitdc  11559  exmidunben  12381  dedekindeulemuub  13389  dedekindeulemlu  13393  dedekindicclemuub  13398  dedekindicclemlu  13402  ivthinclemdisj  13412  exmidsbthrlem  14054
  Copyright terms: Public domain W3C validator