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 4010 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | mpbi 233 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) |
4 | 3 | simpri 489 | 1 ⊢ ¬ 𝐵 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ⊆ wss 3875 ⊊ wpss 3876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-v 3417 df-in 3882 df-ss 3892 df-pss 3894 |
This theorem is referenced by: nsssmfmbf 44001 |
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