Theorem disjeq0 4407

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 4164	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
2		inidm 4178	. . . . . 6 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1, 2	eqtrdi 2786	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
4	3	eqeq1d 2737	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅))
5		eqtr 2755	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
6		simpr 484	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐵 = ∅)
7	5, 6	jca 511	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅))
8	7	ex 412	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
9	4, 8	sylbid 240	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
10	9	com12 32	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		eqtr3 2757	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
12	10, 11	impbid1 225	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∩ cin 3899  ∅c0 4284

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-in 3907

This theorem is referenced by:  epnsym  9520