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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 4245 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | dfss4 4275 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | mpbi 230 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∩ 𝐵) |
4 | difin 4278 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
5 | 4 | difeq2i 4133 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ (𝐴 ∩ 𝐵))) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
6 | 3, 5 | eqtr3i 2765 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 |
This theorem is referenced by: indif 4286 cnvin 6167 imain 6653 resin 6871 elcls 23097 cmmbl 25583 mbfeqalem2 25691 itg1addlem4 25748 itg1addlem4OLD 25749 itg1addlem5 25750 suppovss 32696 inelsiga 34116 inelros 34154 topdifinffinlem 37330 poimirlem9 37616 mblfinlem4 37647 ismblfin 37648 cnambfre 37655 stoweidlem50 46006 saliinclf 46282 sge0fodjrnlem 46372 meadjiunlem 46421 caragendifcl 46470 |
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