Theorem unineq 2226

Description: Infer 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 640	. . . . . . 7 |- (x e. C -> (x e. A <-> (x e. A /\ x e. C)))
2		iba 640	. . . . . . 7 |- (x e. C -> (x e. B <-> (x e. B /\ x e. C)))
3	1, 2	bibi12d 627	. . . . . 6 |- (x e. C -> ((x e. A <-> x e. B) <-> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C))))
4		eleq2 1511	. . . . . . 7 |- ((A i^i C) = (B i^i C) -> (x e. (A i^i C) <-> x e. (B i^i C)))
5		elin 2178	. . . . . . 7 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
6		elin 2178	. . . . . . 7 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
7	4, 5, 6	3bitr3g 552	. . . . . 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 210	. . . . 5 |- (x e. C -> ((A i^i C) = (B i^i C) -> (x e. A <-> x e. B)))
9	8	adantld 390	. . . 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 732	. . . . . . 7 |- (-. x e. C -> (x e. A <-> (x e. C \/ x e. A)))
11		biorf 732	. . . . . . 7 |- (-. x e. C -> (x e. B <-> (x e. C \/ x e. B)))
12	10, 11	bibi12d 627	. . . . . 6 |- (-. x e. C -> ((x e. A <-> x e. B) <-> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B))))
13		uncom 2147	. . . . . . . . 9 |- (A u. C) = (C u. A)
14		uncom 2147	. . . . . . . . 9 |- (B u. C) = (C u. B)
15	13, 14	eqeq12i 1464	. . . . . . . 8 |- ((A u. C) = (B u. C) <-> (C u. A) = (C u. B))
16		eleq2 1511	. . . . . . . 8 |- ((C u. A) = (C u. B) -> (x e. (C u. A) <-> x e. (C u. B)))
17	15, 16	sylbi 199	. . . . . . 7 |- ((A u. C) = (B u. C) -> (x e. (C u. A) <-> x e. (C u. B)))
18		elun 2144	. . . . . . 7 |- (x e. (C u. A) <-> (x e. C \/ x e. A))
19		elun 2144	. . . . . . 7 |- (x e. (C u. B) <-> (x e. C \/ x e. B))
20	17, 18, 19	3bitr3g 552	. . . . . 6 |- ((A u. C) = (B u. C) -> ((x e. C \/ x e. A) <-> (x e. C \/ x e. B)))
21	12, 20	syl5bir 210	. . . . 5 |- (-. x e. C -> ((A u. C) = (B u. C) -> (x e. A <-> x e. B)))
22	21	adantrd 391	. . . 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 126	. . 3 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> (x e. A <-> x e. B))
24	23	eqrdv 1450	. 2 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) -> A = B)
25		uneq1 2148	. . 3 |- (A = B -> (A u. C) = (B u. C))
26		ineq1 2181	. . 3 |- (A = B -> (A i^i C) = (B i^i C))
27	25, 26	jca 288	. 2 |- (A = B -> ((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)))
28	24, 27	impbi 157	1 |- (((A u. C) = (B u. C) /\ (A i^i C) = (B i^i C)) <-> A = B)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 1099   e. wcel 1105   u. cun 2016   i^i cin 2017

This theorem is referenced by:  mapdom2 4426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-un 2021  df-in 2022

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