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Mirrors > Home > MPE Home > Th. List > indif | Structured version Visualization version GIF version |
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 4099 | . 2 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵))) | |
2 | dfin4 4099 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (𝐴 ∖ 𝐵)) | |
3 | 2 | difeq2i 3954 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∖ (𝐴 ∖ 𝐵))) |
4 | difin 4093 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
5 | 1, 3, 4 | 3eqtr2i 2855 | 1 ⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∖ cdif 3795 ∩ cin 3797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-in 3805 df-ss 3812 |
This theorem is referenced by: resdif 6402 kmlem11 9304 psgndiflemB 20313 |
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