Theorem uneqin 4233

Description: Equality of union and intersection is equivalent to equality of the 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 3967	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
2		unss 4187	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) ↔ (𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵))
3		ssin 4134	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝐴 ∩ 𝐵))
4		sstr 3917	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
5	3, 4	sylbir 238	. . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∩ 𝐵) → 𝐴 ⊆ 𝐵)
6		ssin 4134	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) ↔ 𝐵 ⊆ (𝐴 ∩ 𝐵))
7		simpl 488	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
8	6, 7	sylbir 238	. . . . . 6 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
9	5, 8	anim12i 625	. . . . 5 ⊢ ((𝐴 ⊆ (𝐴 ∩ 𝐵) ∧ 𝐵 ⊆ (𝐴 ∩ 𝐵)) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
10	2, 9	sylbir 238	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ⊆ (𝐴 ∩ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
11	1, 10	syl 18	. . 3 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
12		eqss 3924	. . 3 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
13	11, 12	sylibr 237	. 2 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) → 𝐴 = 𝐵)
14		unidm 4155	. . . 4 ⊢ (𝐴 ∪ 𝐴) = 𝐴
15		inidm 4122	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16	14, 15	eqtr4i 2786	. . 3 ⊢ (𝐴 ∪ 𝐴) = (𝐴 ∩ 𝐴)
17		uneq2 4160	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
18		ineq2 4110	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
19	16, 17, 18	3eqtr3a 2819	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵))
20	13, 19	impbii 212	1 ⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-un 4150

This theorem is used by:  uniintsn  4944