Theorem uneqin 2845

Description: Equality of union and intersection implies equality of their arguments. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
uneqin	|- ((A u. B) = (A i^i B) <-> A = B)

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 2665	. . . 4 |- ((A u. B) = (A i^i B) -> (A u. B) C_ (A i^i B))
2		unss 2780	. . . . 5 |- ((A C_ (A i^i B) /\ B C_ (A i^i B)) <-> (A u. B) C_ (A i^i B))
3		ssin 2814	. . . . . . 7 |- ((A C_ A /\ A C_ B) <-> A C_ (A i^i B))
4		sstr 2625	. . . . . . 7 |- ((A C_ A /\ A C_ B) -> A C_ B)
5	3, 4	sylbir 218	. . . . . 6 |- (A C_ (A i^i B) -> A C_ B)
6		ssin 2814	. . . . . . 7 |- ((B C_ A /\ B C_ B) <-> B C_ (A i^i B))
7		simpl 346	. . . . . . 7 |- ((B C_ A /\ B C_ B) -> B C_ A)
8	6, 7	sylbir 218	. . . . . 6 |- (B C_ (A i^i B) -> B C_ A)
9	5, 8	anim12i 360	. . . . 5 |- ((A C_ (A i^i B) /\ B C_ (A i^i B)) -> (A C_ B /\ B C_ A))
10	2, 9	sylbir 218	. . . 4 |- ((A u. B) C_ (A i^i B) -> (A C_ B /\ B C_ A))
11	1, 10	syl 12	. . 3 |- ((A u. B) = (A i^i B) -> (A C_ B /\ B C_ A))
12		eqss 2631	. . 3 |- (A = B <-> (A C_ B /\ B C_ A))
13	11, 12	sylibr 217	. 2 |- ((A u. B) = (A i^i B) -> A = B)
14		uneq2 2749	. . 3 |- (A = B -> (A u. A) = (A u. B))
15		ineq2 2790	. . . 4 |- (A = B -> (A i^i A) = (A i^i B))
16		unidm 2743	. . . . 5 |- (A u. A) = A
17		inidm 2803	. . . . 5 |- (A i^i A) = A
18	16, 17	eqtr4i 1911	. . . 4 |- (A u. A) = (A i^i A)
19	15, 18	syl5eq 1940	. . 3 |- (A = B -> (A u. A) = (A i^i B))
20	14, 19	eqtr3d 1927	. 2 |- (A = B -> (A u. B) = (A i^i B))
21	13, 20	impbii 174	1 |- ((A u. B) = (A i^i B) <-> A = B)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   u. cun 2591   i^i cin 2592   C_ wss 2593

This theorem is referenced by:  uniintsn 3253

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-in 2603  df-ss 2605

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