Theorem dfin3 3736

Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfin3

Proof of Theorem dfin3

Step	Hyp	Ref	Expression
1		ddif 3635	. 2
2		dfun2 3732	. . . 4
3		ddif 3635	. . . . . 6
4	3	difeq1i 3617	. . . . 5
5	4	difeq2i 3618	. . . 4
6	2, 5	eqtri 2486	. . 3
7	6	difeq2i 3618	. 2
8		dfin2 3733	. 2
9	1, 7, 8	3eqtr4ri 2497	1

Syntax hints:  =wceq 1395  cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474

This theorem is referenced by:  difindi  3751

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-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482