Theorem inundif 4431

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 3917	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3911	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 914	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1053	. . 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 4108	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898   ∪ cun 3899   ∩ cin 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-un 3906  df-in 3908

This theorem is referenced by:  iunxdif3  5050  partfun  6639  resasplit  6704  fresaun  6705  fresaunres2  6706  ixpfi2  9250  hashun3  14307  prmreclem2  16845  mvdco  19374  sylow2a  19548  ablfac1eu  20004  basdif0  22897  neitr  23124  cmpfi  23352  ptbasfi  23525  ptcnplem  23565  fin1aufil  23876  ismbl2  25484  volinun  25503  voliunlem2  25508  mbfeqalem2  25599  itg2cnlem2  25719  dvres2lem  25867  indifundif  32599  imadifxp  32676  ofpreima2  32744  resf1o  32809  indsumin  32943  gsummptres  33135  tocyccntz  33226  measun  34368  measunl  34373  inelcarsg  34468  carsgclctun  34478  sibfof  34497  probdif  34577  hgt750lemd  34805  mthmpps  35776  clcnvlem  43864  radcnvrat  44555  sumnnodd  45876  ovolsplit  46232  omelesplit  46762  ovnsplit  46892