Theorem disj1 4412

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	⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem disj1

Step	Hyp	Ref	Expression
1		disj 4410	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		df-ral 3080	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
3	1, 2	bitri 278	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-dif 3908  df-in 3912  df-nul 4287

This theorem is referenced by:  reldisj  4413  disj3  4414  undif4  4427  disjsn  4677  funun  6582  zfregs2  9698  dfac5lem4  10106  isf32lem9  10340  fzodisj  13718  fzodisjsn  13722  inpr0  32878  bnj1280  35408  axregszf  35542  ecin0  39001  zfregs2VD  45549  dfac5prim  45699  permac8prim  45723