Theorem inass 4158

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 468	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
2		elin 3901	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
3	2	anbi2i 624	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4	1, 3	bitr4i 278	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)))
5		elin 3901	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
6	5	anbi1i 625	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶))
7		elin 3901	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶)))
8	4, 6, 7	3bitr4i 303	. 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶)))
9	8	ineqri 4143	1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3429  df-in 3892

This theorem is referenced by:  in12  4159  in32  4160  in4  4164  indif2  4211  difun1  4229  dfrab3ss  4253  dfif4  4472  resres  5946  inres  5951  imainrect  6134  cnvrescnv  6148  predidm  6279  onfr  6351  fresaun  6700  fresaunres2  6701  fimacnvinrn2  7013  epfrs  9641  incexclem  15790  sadeq  16430  smuval2  16440  smumul  16451  ressinbas  17204  ressress  17206  resscatc  18065  sylow2a  19583  ablfac1eu  20039  ressmplbas2  21994  restco  23117  restopnb  23128  kgeni  23490  hausdiag  23598  fclsrest  23977  clsocv  25205  itg2cnlem2  25717  rplogsum  27478  chjassi  31545  pjoml2i  31644  cmcmlem  31650  cmbr3i  31659  fh1  31677  fh2  31678  pj3lem1  32265  dmdbr5  32367  mdslmd3i  32391  mdexchi  32394  atabsi  32460  dmdbr6ati  32482  prsss  34048  inelcarsg  34443  carsgclctunlem1  34449  msrid  35715  dfttc4  36700  redundss3  39021  refrelsredund4  39025  dfpetparts2  39281  dfpeters2  39283  osumcllem9N  40398  dihmeetbclemN  41738  dihmeetlem11N  41751  wfac8prim  45417  inabs3  45475  uzinico2  45979  caragenuncllem  46928  resinsn  49335  restclsseplem  49378