Theorem uneqin 3506

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 3323	. . . 4 ⊢ ((A ∪ B) = (A ∩ B) → (A ∪ B) ⊆ (A ∩ B))
2		unss 3437	. . . . 5 ⊢ ((A ⊆ (A ∩ B) ∧ B ⊆ (A ∩ B)) ↔ (A ∪ B) ⊆ (A ∩ B))
3		ssin 3477	. . . . . . 7 ⊢ ((A ⊆ A ∧ A ⊆ B) ↔ A ⊆ (A ∩ B))
4		sstr 3280	. . . . . . 7 ⊢ ((A ⊆ A ∧ A ⊆ B) → A ⊆ B)
5	3, 4	sylbir 204	. . . . . 6 ⊢ (A ⊆ (A ∩ B) → A ⊆ B)
6		ssin 3477	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) ↔ B ⊆ (A ∩ B))
7		simpl 443	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) → B ⊆ A)
8	6, 7	sylbir 204	. . . . . 6 ⊢ (B ⊆ (A ∩ B) → B ⊆ A)
9	5, 8	anim12i 549	. . . . 5 ⊢ ((A ⊆ (A ∩ B) ∧ B ⊆ (A ∩ B)) → (A ⊆ B ∧ B ⊆ A))
10	2, 9	sylbir 204	. . . 4 ⊢ ((A ∪ B) ⊆ (A ∩ B) → (A ⊆ B ∧ B ⊆ A))
11	1, 10	syl 15	. . 3 ⊢ ((A ∪ B) = (A ∩ B) → (A ⊆ B ∧ B ⊆ A))
12		eqss 3287	. . 3 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
13	11, 12	sylibr 203	. 2 ⊢ ((A ∪ B) = (A ∩ B) → A = B)
14		unidm 3407	. . . 4 ⊢ (A ∪ A) = A
15		inidm 3464	. . . 4 ⊢ (A ∩ A) = A
16	14, 15	eqtr4i 2376	. . 3 ⊢ (A ∪ A) = (A ∩ A)
17		uneq2 3412	. . 3 ⊢ (A = B → (A ∪ A) = (A ∪ B))
18		ineq2 3451	. . 3 ⊢ (A = B → (A ∩ A) = (A ∩ B))
19	16, 17, 18	3eqtr3a 2409	. 2 ⊢ (A = B → (A ∪ B) = (A ∩ B))
20	13, 19	impbii 180	1 ⊢ ((A ∪ B) = (A ∩ B) ↔ A = B)

Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259

This theorem is referenced by:  uniintsn  3963