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Theorem disjeq0 4378
Description: Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
disjeq0 ((𝐴𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Proof of Theorem disjeq0
StepHypRef Expression
1 ineq1 4156 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐵𝐵))
2 inidm 4170 . . . . . 6 (𝐵𝐵) = 𝐵
31, 2syl6eq 2872 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
43eqeq1d 2823 . . . 4 (𝐴 = 𝐵 → ((𝐴𝐵) = ∅ ↔ 𝐵 = ∅))
5 eqtr 2841 . . . . . 6 ((𝐴 = 𝐵𝐵 = ∅) → 𝐴 = ∅)
6 simpr 488 . . . . . 6 ((𝐴 = 𝐵𝐵 = ∅) → 𝐵 = ∅)
75, 6jca 515 . . . . 5 ((𝐴 = 𝐵𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅))
87ex 416 . . . 4 (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
94, 8sylbid 243 . . 3 (𝐴 = 𝐵 → ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
109com12 32 . 2 ((𝐴𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅)))
11 eqtr3 2843 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
1210, 11impbid1 228 1 ((𝐴𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  cin 3909  c0 4266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-rab 3135  df-v 3473  df-in 3917
This theorem is referenced by:  epnsym  9048
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