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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pssnssi | Structured version Visualization version GIF version |
Description: A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pssnssi.1 | ⊢ 𝐴 ⊊ 𝐵 |
Ref | Expression |
---|---|
pssnssi | ⊢ ¬ 𝐵 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssnssi.1 | . . 3 ⊢ 𝐴 ⊊ 𝐵 | |
2 | dfpss3 4081 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | mpbi 229 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) |
4 | 3 | simpri 485 | 1 ⊢ ¬ 𝐵 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ⊆ wss 3943 ⊊ wpss 3944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-in 3950 df-ss 3960 df-pss 3962 |
This theorem is referenced by: nsssmfmbf 46049 |
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