Theorem disj1 4019

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 4017	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
2		df-ral 2917	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
3	1, 2	bitri 264	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  reldisj  4020  disj3  4021  undif4  4035  disjsn  4246  funun  5932  zfregs2  8609  dfac5lem4  8949  isf32lem9  9183  fzodisj  12502  fzodisjsn  12505  bnj1280  31088  ecin0  34117  zfregs2VD  39076