Theorem uneqin 3182

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 2991	. . . 4 ⊢ ((A ∪ B) = (A ∩ B) → (A ∪ B) ⊆ (A ∩ B))
2		unss 3111	. . . . 5 ⊢ ((A ⊆ (A ∩ B) ∧ B ⊆ (A ∩ B)) ↔ (A ∪ B) ⊆ (A ∩ B))
3		ssin 3153	. . . . . . 7 ⊢ ((A ⊆ A ∧ A ⊆ B) ↔ A ⊆ (A ∩ B))
4		sstr 2947	. . . . . . 7 ⊢ ((A ⊆ A ∧ A ⊆ B) → A ⊆ B)
5	3, 4	sylbir 125	. . . . . 6 ⊢ (A ⊆ (A ∩ B) → A ⊆ B)
6		ssin 3153	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) ↔ B ⊆ (A ∩ B))
7		simpl 102	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) → B ⊆ A)
8	6, 7	sylbir 125	. . . . . 6 ⊢ (B ⊆ (A ∩ B) → B ⊆ A)
9	5, 8	anim12i 321	. . . . 5 ⊢ ((A ⊆ (A ∩ B) ∧ B ⊆ (A ∩ B)) → (A ⊆ B ∧ B ⊆ A))
10	2, 9	sylbir 125	. . . 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 2954	. . 3 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
13	11, 12	sylibr 137	. 2 ⊢ ((A ∪ B) = (A ∩ B) → A = B)
14		unidm 3080	. . . 4 ⊢ (A ∪ A) = A
15		inidm 3140	. . . 4 ⊢ (A ∩ A) = A
16	14, 15	eqtr4i 2060	. . 3 ⊢ (A ∪ A) = (A ∩ A)
17		uneq2 3085	. . 3 ⊢ (A = B → (A ∪ A) = (A ∪ B))
18		ineq2 3126	. . 3 ⊢ (A = B → (A ∩ A) = (A ∩ B))
19	16, 17, 18	3eqtr3a 2093	. 2 ⊢ (A = B → (A ∪ B) = (A ∩ B))
20	13, 19	impbii 117	1 ⊢ ((A ∪ B) = (A ∩ B) ↔ A = B)

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925

This theorem is referenced by: (None)