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Mirrors > Home > MPE Home > Th. List > inass | Structured version Visualization version GIF version |
Description: Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) |
Ref | Expression |
---|---|
inass | ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 471 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))) | |
2 | elin 4169 | . . . . 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 280 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶))) |
5 | elin 4169 | . . . 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 4169 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 ∩ 𝐶))) | |
8 | 4, 6, 7 | 3bitr4i 305 | . 2 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑥 ∈ 𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 ∩ 𝐶))) |
9 | 8 | ineqri 4180 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 |
This theorem is referenced by: in12 4197 in32 4198 in4 4202 indif2 4247 difun1 4264 dfrab3ss 4281 dfif4 4482 resres 5866 inres 5871 imainrect 6038 cnvrescnv 6052 predidm 6170 onfr 6230 fresaun 6549 fresaunres2 6550 fimacnvinrn2 6841 epfrs 9173 incexclem 15191 sadeq 15821 smuval2 15831 smumul 15842 ressinbas 16560 ressress 16562 resscatc 17365 sylow2a 18744 ablfac1eu 19195 ressmplbas2 20236 restco 21772 restopnb 21783 kgeni 22145 hausdiag 22253 fclsrest 22632 clsocv 23853 itg2cnlem2 24363 rplogsum 26103 chjassi 29263 pjoml2i 29362 cmcmlem 29368 cmbr3i 29377 fh1 29395 fh2 29396 pj3lem1 29983 dmdbr5 30085 mdslmd3i 30109 mdexchi 30112 atabsi 30178 dmdbr6ati 30200 prsss 31159 inelcarsg 31569 carsgclctunlem1 31575 msrid 32792 redundss3 35878 refrelsredund4 35882 osumcllem9N 37115 dihmeetbclemN 38455 dihmeetlem11N 38468 inabs3 41338 uzinico2 41858 caragenuncllem 42814 |
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