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Mirrors > Home > MPE Home > Th. List > nssinpss | Structured version Visualization version GIF version |
Description: Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
nssinpss | ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4229 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
2 | 1 | biantrur 530 | . 2 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴)) |
3 | df-ss 3964 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
4 | 3 | necon3bbii 2985 | . 2 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ≠ 𝐴) |
5 | df-pss 3966 | . 2 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴)) | |
6 | 2, 4, 5 | 3bitr4i 303 | 1 ⊢ (¬ 𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ≠ wne 2937 ∩ cin 3946 ⊆ wss 3947 ⊊ wpss 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-v 3473 df-in 3954 df-ss 3964 df-pss 3966 |
This theorem is referenced by: fbfinnfr 23744 chrelat2i 32174 |
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