Theorem uneqin 3507

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 u. B) = (A i^i B) <-> A = B)

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 3324	. . . 4 |- ((A u. B) = (A i^i B) -> (A u. B) C_ (A i^i B))
2		unss 3438	. . . . 5 |- ((A C_ (A i^i B) /\ B C_ (A i^i B)) <-> (A u. B) C_ (A i^i B))
3		ssin 3478	. . . . . . 7 |- ((A C_ A /\ A C_ B) <-> A C_ (A i^i B))
4		sstr 3281	. . . . . . 7 |- ((A C_ A /\ A C_ B) -> A C_ B)
5	3, 4	sylbir 204	. . . . . 6 |- (A C_ (A i^i B) -> A C_ B)
6		ssin 3478	. . . . . . 7 |- ((B C_ A /\ B C_ B) <-> B C_ (A i^i B))
7		simpl 443	. . . . . . 7 |- ((B C_ A /\ B C_ B) -> B C_ A)
8	6, 7	sylbir 204	. . . . . 6 |- (B C_ (A i^i B) -> B C_ A)
9	5, 8	anim12i 549	. . . . 5 |- ((A C_ (A i^i B) /\ B C_ (A i^i B)) -> (A C_ B /\ B C_ A))
10	2, 9	sylbir 204	. . . 4 |- ((A u. B) C_ (A i^i B) -> (A C_ B /\ B C_ A))
11	1, 10	syl 15	. . 3 |- ((A u. B) = (A i^i B) -> (A C_ B /\ B C_ A))
12		eqss 3288	. . 3 |- (A = B <-> (A C_ B /\ B C_ A))
13	11, 12	sylibr 203	. 2 |- ((A u. B) = (A i^i B) -> A = B)
14		unidm 3408	. . . 4 |- (A u. A) = A
15		inidm 3465	. . . 4 |- (A i^i A) = A
16	14, 15	eqtr4i 2376	. . 3 |- (A u. A) = (A i^i A)
17		uneq2 3413	. . 3 |- (A = B -> (A u. A) = (A u. B))
18		ineq2 3452	. . 3 |- (A = B -> (A i^i A) = (A i^i B))
19	16, 17, 18	3eqtr3a 2409	. 2 |- (A = B -> (A u. B) = (A i^i B))
20	13, 19	impbii 180	1 |- ((A u. B) = (A i^i B) <-> A = B)

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642   u. cun 3208   i^i cin 3209   C_ wss 3258

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  uniintsn  3964