Theorem inundif 3998

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 3758	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3550	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 542	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 995	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	3, 4	bitr4i 266	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴)
6	5	uneqri 3717	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ 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-v 3175  df-dif 3543  df-un 3545  df-in 3547

This theorem is referenced by:  iunxdif3  4542  resasplit  5987  fresaun  5988  fresaunres2  5989  ixpfi2  8147  hashun3  13034  prmreclem2  15459  mvdco  17688  sylow2a  17857  ablfac1eu  18295  basdif0  20568  neitr  20794  cmpfi  21021  ptbasfi  21194  ptcnplem  21234  fin1aufil  21546  ismbl2  23102  volinun  23121  voliunlem2  23126  mbfeqalem  23215  itg2cnlem2  23335  dvres2lem  23480  indifundif  28740  imadifxp  28796  ofpreima2  28849  partfun  28858  resf1o  28893  gsummptres  29115  measun  29601  measunl  29606  inelcarsg  29700  carsgclctun  29710  sibfof  29729  probdif  29809  mthmpps  30733  clcnvlem  36949  radcnvrat  37535  sumnnodd  38697  ovolsplit  38881  omelesplit  39408  ovnsplit  39538