Theorem uneqin 3458

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 3281	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
2		unss 3381	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) ↔ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
3		ssin 3429	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝐴 ∩ 𝐵))
4		sstr 3235	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	sylbir 135	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∩ 𝐵) → 𝐴 ⊆ 𝐵)
6		ssin 3429	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
7		simpl 109	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
8	6, 7	sylbir 135	. . . . . 6 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
9	5, 8	anim12i 338	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
10	2, 9	sylbir 135	. . . 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 3242	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
13	11, 12	sylibr 134	. 2 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → 𝐴 = 𝐵)
14		unidm 3350	. . . 4 ⊢ (𝐴 ∪ 𝐴) = 𝐴
15		inidm 3416	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16	14, 15	eqtr4i 2255	. . 3 ⊢ (𝐴 ∪ 𝐴) = (𝐴 ∩ 𝐴)
17		uneq2 3355	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
18		ineq2 3402	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
19	16, 17, 18	3eqtr3a 2288	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵))
20	13, 19	impbii 126	1 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by: (None)