Theorem disj1 4404

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 4402	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		df-ral 3052	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
3	1, 2	bitri 275	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-dif 3904  df-in 3908  df-nul 4286

This theorem is referenced by:  reldisj  4405  disj3  4406  undif4  4419  disjsn  4668  funun  6538  zfregs2  9642  dfac5lem4  10036  dfac5lem4OLD  10038  isf32lem9  10271  fzodisj  13609  fzodisjsn  13613  inpr0  32607  bnj1280  35176  axregszf  35285  ecin0  38545  zfregs2VD  45081  dfac5prim  45231  permac8prim  45255