Theorem inundif 4419

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 3905	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3899	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 915	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1054	. . 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 4096	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-un 3894  df-in 3896

This theorem is referenced by:  iunxdif3  5037  partfun  6645  resasplit  6710  fresaun  6711  fresaunres2  6712  ixpfi2  9260  hashun3  14346  prmreclem2  16888  mvdco  19420  sylow2a  19594  ablfac1eu  20050  basdif0  22918  neitr  23145  cmpfi  23373  ptbasfi  23546  ptcnplem  23586  fin1aufil  23897  ismbl2  25494  volinun  25513  voliunlem2  25518  mbfeqalem2  25609  itg2cnlem2  25729  dvres2lem  25877  indifundif  32594  imadifxp  32671  ofpreima2  32739  resf1o  32803  indsumin  32921  gsummptres  33113  tocyccntz  33205  measun  34355  measunl  34360  inelcarsg  34455  carsgclctun  34465  sibfof  34484  probdif  34564  hgt750lemd  34792  mthmpps  35764  clcnvlem  44050  radcnvrat  44741  sumnnodd  46060  ovolsplit  46416  omelesplit  46946  ovnsplit  47076