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Mirrors > Home > MPE Home > Th. List > dfin4 | Structured version Visualization version GIF version |
Description: Alternate definition of the intersection of two classes. Exercise 4.10(q) of [Mendelson] p. 231. (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
dfin4 | ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4155 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | dfss4 4185 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | mpbi 233 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵) |
4 | difin 4188 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
5 | 4 | difeq2i 4047 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
6 | 3, 5 | eqtr3i 2823 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 |
This theorem is referenced by: indif 4196 cnvin 5970 imain 6409 resin 6611 elcls 21678 cmmbl 24138 mbfeqalem2 24246 itg1addlem4 24303 itg1addlem5 24304 suppovss 30443 inelsiga 31504 inelros 31542 topdifinffinlem 34764 poimirlem9 35066 mblfinlem4 35097 ismblfin 35098 cnambfre 35105 stoweidlem50 42692 saliincl 42967 sge0fodjrnlem 43055 meadjiunlem 43104 caragendifcl 43153 |
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