Theorem undif1 3995

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif 3992). 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 3835	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ 𝐵) = ((𝐴 ∪ 𝐵) ∩ ((V ∖ 𝐵) ∪ 𝐵))
2		invdif 3827	. . 3 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵)
3	2	uneq1i 3725	. 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∪ 𝐵) = ((𝐴 ∖ 𝐵) ∪ 𝐵)
4		uncom 3719	. . . . 5 ⊢ ((V ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (V ∖ 𝐵))
5		unvdif 3994	. . . . 5 ⊢ (𝐵 ∪ (V ∖ 𝐵)) = V
6	4, 5	eqtri 2632	. . . 4 ⊢ ((V ∖ 𝐵) ∪ 𝐵) = V
7	6	ineq2i 3773	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∩ ((V ∖ 𝐵) ∪ 𝐵)) = ((𝐴 ∪ 𝐵) ∩ V)
8		inv1 3922	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∩ V) = (𝐴 ∪ 𝐵)
9	7, 8	eqtri 2632	. 2 ⊢ ((𝐴 ∪ 𝐵) ∩ ((V ∖ 𝐵) ∪ 𝐵)) = (𝐴 ∪ 𝐵)
10	1, 3, 9	3eqtr3i 2640	1 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)

Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875

This theorem is referenced by:  undif2  3996  unidif0  4764  pwundif  4945  sofld  5500  fresaun  5988  ralxpmap  7793  enp1ilem  8079  difinf  8115  pwfilem  8143  infdif  8914  fin23lem11  9022  fin1a2lem13  9117  axcclem  9162  ttukeylem1  9214  ttukeylem7  9220  fpwwe2lem13  9343  hashbclem  13093  incexclem  14407  ramub1lem1  15568  ramub1lem2  15569  isstruct2  15704  setsdm  15724  mrieqvlemd  16112  mreexmrid  16126  islbs3  18976  lbsextlem4  18982  basdif0  20568  bwth  21023  locfincmp  21139  cldsubg  21724  nulmbl2  23111  volinun  23121  limcdif  23446  ellimc2  23447  limcmpt2  23454  dvreslem  23479  dvaddbr  23507  dvmulbr  23508  lhop  23583  plyeq0  23771  rlimcnp  24492  difeq  28739  ffsrn  28892  esumpad2  29445  measunl  29606  subfacp1lem1  30415  cvmscld  30509  compne  37665  stoweidlem44  38937