Theorem unineq 1680

Description: Deduce equality from equalities of union and intersection. Exercise 20 of [Enderton] p. 32 and its converse.

Assertion

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

Proof of Theorem unineq

Step	Hyp	Ref	Expression
1		iba 486	. . . . . . 7 |- (x e. C -> (x e. A <-> (x e. A /\ x e. C)))
2		iba 486	. . . . . . 7 |- (x e. C -> (x e. B <-> (x e. B /\ x e. C)))
3	1, 2	bibi12d 477	. . . . . 6 |- (x e. C -> ((x e. A <-> x e. B) <-> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C))))
4		eleq2 1150	. . . . . . 7 |- ((A i^i C) = (B i^i C) -> (x e. (A i^i C) <-> x e. (B i^i C)))
5		elin 1635	. . . . . . 7 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
6		elin 1635	. . . . . . 7 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
7	4, 5, 6	3bitr3g 427	. . . . . 6 |- ((A i^i C) = (B i^i C) -> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C)))
8	3, 7	syl5bir 184	. . . . 5 |- (x e. C -> ((A i^i C) = (B i^i C) -> (x e. A <-> x e. B)))
9	8	adantld 307	. . . 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		biorf 551	. . . . . . 7 |- (-. x e. C -> (x e. A <-> (x e. C \/ x e. A)))
11		biorf 551	. . . . . . 7 |- (-. x e. C -> (x e. B <-> (x e. C \/ x e. B)))
12	10, 11	bibi12d 477	. . . . . 6 |- (-. x e. C -> ((x e. A <-> x e. B) <-> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B))))
13		uncom 1604	. . . . . . . . 9 |- (A u. C) = (C u. A)
14		uncom 1604	. . . . . . . . 9 |- (B u. C) = (C u. B)
15	13, 14	cleq12i 1114	. . . . . . . 8 |- ((A u. C) = (B u. C) <-> (C u. A) = (C u. B))
16		eleq2 1150	. . . . . . . 8 |- ((C u. A) = (C u. B) -> (x e. (C u. A) <-> x e. (C u. B)))
17	15, 16	sylbi 174	. . . . . . 7 |- ((A u. C) = (B u. C) -> (x e. (C u. A) <-> x e. (C u. B)))
18		elun 1601	. . . . . . 7 |- (x e. (C u. A) <-> (x e. C \/ x e. A))
19		elun 1601	. . . . . . 7 |- (x e. (C u. B) <-> (x e. C \/ x e. B))
20	17, 18, 19	3bitr3g 427	. . . . . 6 |- ((A u. C) = (B u. C) -> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B)))
21	12, 20	syl5bir 184	. . . . 5 |- (-. x e. C -> ((A u. C) = (B u. C) -> (x e. A <-> x e. B)))
22	21	adantrd 308	. . . 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 110	. . 3 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> (x e. A <-> x e. B))
24	23	cleqrd 1100	. 2 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> A = B)
25		uneq1 1605	. . 3 |- (A = B -> (A u. C) = (B u. C))
26		ineq1 1638	. . 3 |- (A = B -> (A i^i C) = (B i^i C))
27	25, 26	jca 236	. 2 |- (A = B -> ((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)))
28	24, 27	impbi 139	1 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) <-> A = B)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   u. cun 1485   i^i cin 1486

This theorem is referenced by:  mapdom2 3389

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491

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