Theorem disjeq0 4405

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 4181	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
2		inidm 4195	. . . . . 6 ⊢ (𝐵 ∩ 𝐵) = 𝐵
3	1, 2	syl6eq 2872	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
4	3	eqeq1d 2823	. . . 4 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅))
5		eqtr 2841	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
6		simpr 487	. . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → 𝐵 = ∅)
7	5, 6	jca 514	. . . . 5 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∧ 𝐵 = ∅))
8	7	ex 415	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐵 = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
9	4, 8	sylbid 242	. . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅)))
10	9	com12 32	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 → (𝐴 = ∅ ∧ 𝐵 = ∅)))
11		eqtr3 2843	. 2 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = 𝐵)
12	10, 11	impbid1 227	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 = 𝐵 ↔ (𝐴 = ∅ ∧ 𝐵 = ∅)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∩ cin 3935  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-in 3943

This theorem is referenced by:  epnsym  9066