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Theorem disj 3313
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 2990 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
21eqeq1i 2090 . . 3 ((𝐴𝐵) = ∅ ↔ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅)
3 abeq1 2192 . . 3 ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} = ∅ ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
4 imnan 657 . . . . 5 ((𝑥𝐴 → ¬ 𝑥𝐵) ↔ ¬ (𝑥𝐴𝑥𝐵))
5 noel 3273 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 648 . . . . 5 (¬ (𝑥𝐴𝑥𝐵) ↔ ((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅))
74, 6bitr2i 183 . . . 4 (((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ (𝑥𝐴 → ¬ 𝑥𝐵))
87albii 1400 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ↔ 𝑥 ∈ ∅) ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
92, 3, 83bitri 204 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
10 df-ral 2358 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
119, 10bitr4i 185 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  cin 2983  c0 3269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-dif 2986  df-in 2990  df-nul 3270
This theorem is referenced by:  disjr  3314  disj1  3315  disjne  3318  f0rn0  5153  renfdisj  7449  fvinim0ffz  9541  fxnn0nninf  9733
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