Theorem unineq 3506

Description: Infer equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
unineq	|- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) <-> A = B)

Proof of Theorem unineq

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . . . 7 |- ((A i^i C) = (B i^i C) -> (x e. (A i^i C) <-> x e. (B i^i C)))
2		elin 3220	. . . . . . 7 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
3		elin 3220	. . . . . . 7 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
4	1, 2, 3	3bitr3g 278	. . . . . 6 |- ((A i^i C) = (B i^i C) -> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C)))
5		iba 489	. . . . . . 7 |- (x e. C -> (x e. A <-> (x e. A /\ x e. C)))
6		iba 489	. . . . . . 7 |- (x e. C -> (x e. B <-> (x e. B /\ x e. C)))
7	5, 6	bibi12d 312	. . . . . 6 |- (x e. C -> ((x e. A <-> x e. B) <-> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C))))
8	4, 7	syl5ibr 212	. . . . 5 |- (x e. C -> ((A i^i C) = (B i^i C) -> (x e. A <-> x e. B)))
9	8	adantld 453	. . . 4 |- (x e. C -> (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> (x e. A <-> x e. B)))
10		uncom 3409	. . . . . . . . 9 |- (A u. C) = (C u. A)
11		uncom 3409	. . . . . . . . 9 |- (B u. C) = (C u. B)
12	10, 11	eqeq12i 2366	. . . . . . . 8 |- ((A u. C) = (B u. C) <-> (C u. A) = (C u. B))
13		eleq2 2414	. . . . . . . 8 |- ((C u. A) = (C u. B) -> (x e. (C u. A) <-> x e. (C u. B)))
14	12, 13	sylbi 187	. . . . . . 7 |- ((A u. C) = (B u. C) -> (x e. (C u. A) <-> x e. (C u. B)))
15		elun 3221	. . . . . . 7 |- (x e. (C u. A) <-> (x e. C \/ x e. A))
16		elun 3221	. . . . . . 7 |- (x e. (C u. B) <-> (x e. C \/ x e. B))
17	14, 15, 16	3bitr3g 278	. . . . . 6 |- ((A u. C) = (B u. C) -> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B)))
18		biorf 394	. . . . . . 7 |- (-. x e. C -> (x e. A <-> (x e. C \/ x e. A)))
19		biorf 394	. . . . . . 7 |- (-. x e. C -> (x e. B <-> (x e. C \/ x e. B)))
20	18, 19	bibi12d 312	. . . . . 6 |- (-. x e. C -> ((x e. A <-> x e. B) <-> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B))))
21	17, 20	syl5ibr 212	. . . . 5 |- (-. x e. C -> ((A u. C) = (B u. C) -> (x e. A <-> x e. B)))
22	21	adantrd 454	. . . 4 |- (-. x e. C -> (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> (x e. A <-> x e. B)))
23	9, 22	pm2.61i 156	. . 3 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> (x e. A <-> x e. B))
24	23	eqrdv 2351	. 2 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> A = B)
25		uneq1 3412	. . 3 |- (A = B -> (A u. C) = (B u. C))
26		ineq1 3451	. . 3 |- (A = B -> (A i^i C) = (B i^i C))
27	25, 26	jca 518	. 2 |- (A = B -> ((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)))
28	24, 27	impbii 180	1 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) <-> A = B)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   u. cun 3208   i^i cin 3209

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

This theorem is referenced by:  phiall  4619