Theorem undif1 3626

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3623). Theorem 35 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
undif1	⊢ ((A ∖ B) ∪ B) = (A ∪ B)

Proof of Theorem undif1

Step	Hyp	Ref	Expression
1		undir 3505	. 2 ⊢ ((A ∩ (V ∖ B)) ∪ B) = ((A ∪ B) ∩ ((V ∖ B) ∪ B))
2		invdif 3497	. . 3 ⊢ (A ∩ (V ∖ B)) = (A ∖ B)
3	2	uneq1i 3415	. 2 ⊢ ((A ∩ (V ∖ B)) ∪ B) = ((A ∖ B) ∪ B)
4		uncom 3409	. . . . 5 ⊢ ((V ∖ B) ∪ B) = (B ∪ (V ∖ B))
5		undifv 3625	. . . . 5 ⊢ (B ∪ (V ∖ B)) = V
6	4, 5	eqtri 2373	. . . 4 ⊢ ((V ∖ B) ∪ B) = V
7	6	ineq2i 3455	. . 3 ⊢ ((A ∪ B) ∩ ((V ∖ B) ∪ B)) = ((A ∪ B) ∩ V)
8		inv1 3578	. . 3 ⊢ ((A ∪ B) ∩ V) = (A ∪ B)
9	7, 8	eqtri 2373	. 2 ⊢ ((A ∪ B) ∩ ((V ∖ B) ∪ B)) = (A ∪ B)
10	1, 3, 9	3eqtr3i 2381	1 ⊢ ((A ∖ B) ∪ B) = (A ∪ B)

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  undif2  3627  nnsucelrlem4  4428  ssfin  4471  sfinltfin  4536