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 3920	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3914	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 925	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1065	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	3, 4	bitr4i 280	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴)
6	5	uneqri 4109	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-in 3911

This theorem is referenced by:  iunxdif3  5052  partfun  6668  resasplit  6734  fresaun  6735  fresaunres2  6736  ixpfi2  9293  hashun3  14397  prmreclem2  16953  mvdco  19485  sylow2a  19659  ablfac1eu  20115  basdif0  23010  neitr  23237  cmpfi  23465  ptbasfi  23638  ptcnplem  23678  fin1aufil  23989  ismbl2  25586  volinun  25605  voliunlem2  25610  mbfeqalem2  25701  itg2cnlem2  25821  dvres2lem  25969  indifundif  32720  imadifxp  32798  ofpreima2  32865  resf1o  32929  indsumin  33036  gsummptres  33229  tocyccntz  33321  measun  34505  measunl  34510  inelcarsg  34605  carsgclctun  34615  sibfof  34634  probdif  34714  hgt750lemd  34939  mthmpps  35929  clcnvlem  44196  radcnvrat  44887  sumnnodd  46203  ovolsplit  46559  omelesplit  47089  ovnsplit  47219