Theorem inundif 4433

Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
inundif	⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Proof of Theorem inundif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3921	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3915	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 913	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1052	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	3, 4	bitr4i 278	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴)
6	5	uneqri 4106	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 397   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3445  df-dif 3908  df-un 3910  df-in 3912

This theorem is referenced by:  iunxdif3  5050  partfun  6640  resasplit  6704  fresaun  6705  fresaunres2  6706  ixpfi2  9224  hashun3  14208  prmreclem2  16720  mvdco  19154  sylow2a  19325  ablfac1eu  19775  basdif0  22213  neitr  22441  cmpfi  22669  ptbasfi  22842  ptcnplem  22882  fin1aufil  23193  ismbl2  24801  volinun  24820  voliunlem2  24825  mbfeqalem2  24916  itg2cnlem2  25037  dvres2lem  25184  indifundif  31224  imadifxp  31291  ofpreima2  31354  resf1o  31416  gsummptres  31663  tocyccntz  31762  indsumin  32352  measun  32541  measunl  32546  inelcarsg  32642  carsgclctun  32652  sibfof  32671  probdif  32751  hgt750lemd  32992  mthmpps  33907  clcnvlem  41604  radcnvrat  42305  sumnnodd  43559  ovolsplit  43917  omelesplit  44445  ovnsplit  44575