Theorem inass 4179

Description: Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Assertion

Ref	Expression
inass	⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))

Proof of Theorem inass

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 473	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
2		elin 3920	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
3	2	anbi2i 634	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4	1, 3	bitr4i 281	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)))
5		elin 3920	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
6	5	anbi1i 635	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶))
7		elin 3920	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)))
8	4, 6, 7	3bitr4i 306	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶)))
9	8	ineqri 4164	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))

This proof depends on syntax axioms:   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ∩ cin 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911

This theorem is used by:  in12  4180  in32  4181  in4  4185  indif2  4233  difun1  4251  dfrab3ss  4275  dfif4  4502  resres  5990  inres  5995  imainrect  6178  cnvrescnv  6193  predidm  6327  onfr  6400  fresaun  6749  fresaunres2  6750  fimacnvinrn2  7067  epfrs  9698  incexclem  15897  sadeq  16536  smuval2  16546  smumul  16557  ressinbas  17311  ressress  17313  resscatc  18172  sylow2a  19695  ablfac1eu  20151  ressmplbas2  22188  restco  23332  restopnb  23343  kgeni  23705  hausdiag  23813  fclsrest  24192  clsocv  25420  itg2cnlem2  25932  rplogsum  27702  chjassi  31849  pjoml2i  31948  cmcmlem  31954  cmbr3i  31963  fh1  31981  fh2  31982  pj3lem1  32569  dmdbr5  32671  mdslmd3i  32695  mdexchi  32698  atabsi  32764  dmdbr6ati  32786  prsss  34315  inelcarsg  34710  carsgclctunlem1  34716  msrid  36045  dfttc4  37069  redundss3  39389  refrelsredund4  39393  dfpetparts2  39649  dfpeters2  39651  osumcllem9N  40766  dihmeetbclemN  42106  dihmeetlem11N  42119  wfac8prim  45739  inabs3  45804  uzinico2  46305  caragenuncllem  47254  resinsn  49678  restclsseplem  49721