Theorem undif1 2340

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see difdisj 2337). Theorem 35 of [Suppes] p. 29.

Assertion

Ref	Expression
undif1	|- ((A \ B) u. B) = (A u. B)

Proof of Theorem undif1

Step	Hyp	Ref	Expression
1		invdif 2249	. . 3 |- (A i^i (V \ B)) = (A \ B)
2	1	uneq1i 2180	. 2 |- ((A i^i (V \ B)) u. B) = ((A \ B) u. B)
3		undir 2254	. . 3 |- ((A i^i (V \ B)) u. B) = ((A u. B) i^i ((V \ B) u. B))
4		uncom 2176	. . . . 5 |- ((V \ B) u. B) = (B u. (V \ B))
5		undifv 2339	. . . . 5 |- (B u. (V \ B)) = V
6	4, 5	eqtr 1495	. . . 4 |- ((V \ B) u. B) = V
7	6	ineq2i 2214	. . 3 |- ((A u. B) i^i ((V \ B) u. B)) = ((A u. B) i^i V)
8		inv1 2299	. . 3 |- ((A u. B) i^i V) = (A u. B)
9	3, 7, 8	3eqtr 1499	. 2 |- ((A i^i (V \ B)) u. B) = (A u. B)
10	2, 9	eqtr3 1497	1 |- ((A \ B) u. B) = (A u. B)

Syntax hints:   = wceq 956  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046

This theorem is referenced by:  undif2 2341  unidif0 2739  pwfilemOLD 4570  infdif 7568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281

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