Theorem uneqin 3373

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 3196	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
2		unss 3296	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) ↔ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
3		ssin 3344	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝐴 ∩ 𝐵))
4		sstr 3150	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	sylbir 134	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∩ 𝐵) → 𝐴 ⊆ 𝐵)
6		ssin 3344	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
7		simpl 108	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
8	6, 7	sylbir 134	. . . . . 6 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
9	5, 8	anim12i 336	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
10	2, 9	sylbir 134	. . . 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 3157	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
13	11, 12	sylibr 133	. 2 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → 𝐴 = 𝐵)
14		unidm 3265	. . . 4 ⊢ (𝐴 ∪ 𝐴) = 𝐴
15		inidm 3331	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16	14, 15	eqtr4i 2189	. . 3 ⊢ (𝐴 ∪ 𝐴) = (𝐴 ∩ 𝐴)
17		uneq2 3270	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
18		ineq2 3317	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
19	16, 17, 18	3eqtr3a 2223	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵))
20	13, 19	impbii 125	1 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343   ∪ cun 3114   ∩ cin 3115   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129

This theorem is referenced by: (None)