Theorem uneqin 3748

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

Proof of Theorem uneqin

Step	Hyp	Ref	Expression
1		eqimss 3555	. . . 4
2		unss 3677	. . . . 5
3		ssin 3719	. . . . . . 7
4		sstr 3511	. . . . . . 7
5	3, 4	sylbir 213	. . . . . 6
6		ssin 3719	. . . . . . 7
7		simpl 457	. . . . . . 7
8	6, 7	sylbir 213	. . . . . 6
9	5, 8	anim12i 566	. . . . 5
10	2, 9	sylbir 213	. . . 4
11	1, 10	syl 16	. . 3
12		eqss 3518	. . 3
13	11, 12	sylibr 212	. 2
14		unidm 3646	. . . 4
15		inidm 3706	. . . 4
16	14, 15	eqtr4i 2489	. . 3
17		uneq2 3651	. . 3
18		ineq2 3693	. . 3
19	16, 17, 18	3eqtr3a 2522	. 2
20	13, 19	impbii 188	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  u.cun 3473  i^icin 3474  C_wss 3475

This theorem is referenced by:  uniintsn  4324

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  df-ss 3489