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Mirrors > Home > MPE Home > Th. List > difin0 | Structured version Visualization version GIF version |
Description: The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difin0 | ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4156 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | ssdif0 4277 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 |
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 df-nul 4244 |
This theorem is referenced by: volinun 24150 |
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