Theorem disjeq0 4363

Description: Two disjoint sets are equal iff both are empty. (Contributed by AV, 19-Jun-2022.)

Assertion

Ref	Expression
disjeq0	⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Proof of Theorem disjeq0

Step	Hyp	Ref	Expression
1		ineq1 4131	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
2		inidm 4145	. . . . . 6 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1, 2	eqtrdi 2849	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
4	3	eqeq1d 2800	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅))
5		eqtr 2818	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
6		simpr 488	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐵 = ∅)
7	5, 6	jca 515	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅))
8	7	ex 416	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
9	4, 8	sylbid 243	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
10	9	com12 32	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		eqtr3 2820	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
12	10, 11	impbid1 228	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∩ cin 3880  ∅c0 4243

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888

This theorem is referenced by:  epnsym  9056