Theorem dfun2 3732

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3733 and dfss4 3731 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation \ (class difference). (Contributed by NM, 10-Jun-2004.)

Assertion

Ref	Expression
dfun2

Proof of Theorem dfun2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . . . . . 7
2		eldif 3485	. . . . . . 7
3	1, 2	mpbiran 918	. . . . . 6
4	3	anbi1i 695	. . . . 5
5		eldif 3485	. . . . 5
6		ioran 490	. . . . 5
7	4, 5, 6	3bitr4i 277	. . . 4
8	7	con2bii 332	. . 3
9		eldif 3485	. . . 4
10	1, 9	mpbiran 918	. . 3
11	8, 10	bitr4i 252	. 2
12	11	uneqri 3645	1

Syntax hints:  -.wn 3  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  \cdif 3472  u.cun 3473

This theorem is referenced by:  dfun3  3735  dfin3  3736

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-dif 3478  df-un 3480