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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 4207 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | dfss4 4237 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | mpbi 232 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵) |
4 | difin 4240 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
5 | 4 | difeq2i 4098 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
6 | 3, 5 | eqtr3i 2848 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 |
This theorem is referenced by: indif 4248 cnvin 6005 imain 6441 resin 6638 elcls 21683 cmmbl 24137 mbfeqalem2 24245 itg1addlem4 24302 itg1addlem5 24303 suppovss 30428 inelsiga 31396 inelros 31434 topdifinffinlem 34630 poimirlem9 34903 mblfinlem4 34934 ismblfin 34935 cnambfre 34942 stoweidlem50 42342 saliincl 42617 sge0fodjrnlem 42705 meadjiunlem 42754 caragendifcl 42803 |
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