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| Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) | 
| Ref | Expression | 
|---|---|
| indif | ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfin4 4277 | . 2 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵))) | |
| 2 | dfin4 4277 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) | |
| 3 | 2 | difeq2i 4122 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵))) | 
| 4 | difin 4271 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
| 5 | 1, 3, 4 | 3eqtr2i 2770 | 1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∖ cdif 3947 ∩ cin 3949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 | 
| This theorem is referenced by: resdif 6868 kmlem11 10202 psgndiflemB 21619 | 
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