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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 4205 | . 2 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
2 | ssdif0 4322 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅) | |
3 | 1, 2 | mpbi 232 | 1 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-nul 4291 |
This theorem is referenced by: volinun 24146 |
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