Theorem uneqin 3215

Description: Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
uneqin	⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 3051	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
2		unss 3146	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) ↔ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
3		ssin 3188	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝐴 ∩ 𝐵))
4		sstr 3007	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	sylbir 133	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∩ 𝐵) → 𝐴 ⊆ 𝐵)
6		ssin 3188	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
7		simpl 107	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
8	6, 7	sylbir 133	. . . . . 6 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
9	5, 8	anim12i 331	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
10	2, 9	sylbir 133	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
11	1, 10	syl 14	. . 3 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
12		eqss 3014	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
13	11, 12	sylibr 132	. 2 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → 𝐴 = 𝐵)
14		unidm 3115	. . . 4 ⊢ (𝐴 ∪ 𝐴) = 𝐴
15		inidm 3175	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16	14, 15	eqtr4i 2104	. . 3 ⊢ (𝐴 ∪ 𝐴) = (𝐴 ∩ 𝐴)
17		uneq2 3120	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
18		ineq2 3161	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
19	16, 17, 18	3eqtr3a 2137	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵))
20	13, 19	impbii 124	1 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284   ∪ cun 2971   ∩ cin 2972   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986

This theorem is referenced by: (None)