Theorem inundif 4435

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 3915	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		eldif 3909	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	orbi12i 928	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
4		pm4.42 1069	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)))
5	3, 4	bitr4i 281	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ 𝑥 ∈ 𝐴)
6	5	uneqri 4103	1 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-un 3904  df-in 3906

This theorem is used by:  iunxdif3  5055  partfun  6684  resasplit  6750  fresaun  6751  fresaunres2  6752  ixpfi2  9332  hashun3  14521  prmreclem2  17088  mvdco  19652  sylow2a  19826  ablfac1eu  20282  basdif0  23264  neitr  23491  cmpfi  23719  ptbasfi  23893  ptcnplem  23933  fin1aufil  24244  ismbl2  25841  volinun  25860  voliunlem2  25865  mbfeqalem2  25956  itg2cnlem2  26076  dvres2lem  26223  indifundif  33113  imadifxp  33188  ofpreima2  33253  resf1o  33315  indsumin  33421  gsummptres  33606  tocyccntz  33698  measun  34837  measunl  34842  inelcarsg  34936  carsgclctun  34946  sibfof  34965  probdif  35045  hgt750lemd  35270  mthmpps  36326  clcnvlem  44608  radcnvrat  45283  sumnnodd  46611  ovolsplit  46967  omelesplit  47497  ovnsplit  47627