Theorem inundif 4427

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 4169	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3946	. . . 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 4127	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:  ¬ wn 3   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-dif 3939  df-un 3941  df-in 3943

This theorem is referenced by:  iunxdif3  5010  resasplit  6543  fresaun  6544  fresaunres2  6545  ixpfi2  8816  hashun3  13739  prmreclem2  16247  mvdco  18567  sylow2a  18738  ablfac1eu  19189  basdif0  21555  neitr  21782  cmpfi  22010  ptbasfi  22183  ptcnplem  22223  fin1aufil  22534  ismbl2  24122  volinun  24141  voliunlem2  24146  mbfeqalem2  24237  itg2cnlem2  24357  dvres2lem  24502  indifundif  30279  imadifxp  30345  ofpreima2  30405  partfun  30415  resf1o  30460  gsummptres  30685  tocyccntz  30781  indsumin  31276  measun  31465  measunl  31470  inelcarsg  31564  carsgclctun  31574  sibfof  31593  probdif  31673  hgt750lemd  31914  mthmpps  32824  clcnvlem  39976  radcnvrat  40639  sumnnodd  41903  ovolsplit  42266  omelesplit  42793  ovnsplit  42923