Theorem unineq 3747

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

Proof of Theorem unineq

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2530	. . . . . . 7
2		elin 3686	. . . . . . 7
3		elin 3686	. . . . . . 7
4	1, 2, 3	3bitr3g 287	. . . . . 6
5		iba 503	. . . . . . 7
6		iba 503	. . . . . . 7
7	5, 6	bibi12d 321	. . . . . 6
8	4, 7	syl5ibr 221	. . . . 5
9	8	adantld 467	. . . 4
10		uncom 3647	. . . . . . . . 9
11		uncom 3647	. . . . . . . . 9
12	10, 11	eqeq12i 2477	. . . . . . . 8
13		eleq2 2530	. . . . . . . 8
14	12, 13	sylbi 195	. . . . . . 7
15		elun 3644	. . . . . . 7
16		elun 3644	. . . . . . 7
17	14, 15, 16	3bitr3g 287	. . . . . 6
18		biorf 405	. . . . . . 7
19		biorf 405	. . . . . . 7
20	18, 19	bibi12d 321	. . . . . 6
21	17, 20	syl5ibr 221	. . . . 5
22	21	adantrd 468	. . . 4
23	9, 22	pm2.61i 164	. . 3
24	23	eqrdv 2454	. 2
25		uneq1 3650	. . 3
26		ineq1 3692	. . 3
27	25, 26	jca 532	. 2
28	24, 27	impbii 188	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  u.cun 3473  i^icin 3474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482