Theorem inundif 4425

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 4167	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3944	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 911	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1048	. . 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 4125	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 398   ∨ wo 843   = wceq 1531   ∈ wcel 2108   ∖ cdif 3931   ∪ cun 3932   ∩ cin 3933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-un 3939  df-in 3941

This theorem is referenced by:  iunxdif3  5008  resasplit  6541  fresaun  6542  fresaunres2  6543  ixpfi2  8814  hashun3  13737  prmreclem2  16245  mvdco  18565  sylow2a  18736  ablfac1eu  19187  basdif0  21553  neitr  21780  cmpfi  22008  ptbasfi  22181  ptcnplem  22221  fin1aufil  22532  ismbl2  24120  volinun  24139  voliunlem2  24144  mbfeqalem2  24235  itg2cnlem2  24355  dvres2lem  24500  indifundif  30277  imadifxp  30343  ofpreima2  30403  partfun  30413  resf1o  30458  gsummptres  30683  tocyccntz  30779  indsumin  31274  measun  31463  measunl  31468  inelcarsg  31562  carsgclctun  31572  sibfof  31591  probdif  31671  hgt750lemd  31912  mthmpps  32822  clcnvlem  39973  radcnvrat  40636  sumnnodd  41900  ovolsplit  42263  omelesplit  42790  ovnsplit  42920