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Theorem disj1 3270
 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 3268 . 2 ((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
2 df-ral 2311 . 2 (∀𝑥𝐴 ¬ 𝑥𝐵 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
31, 2bitri 173 1 ((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ∩ cin 2916  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-nul 3225 This theorem is referenced by:  reldisj  3271  disj3  3272  undif4  3284  disjsn  3432  funun  4944  fzodisj  9034
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