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 4060 | . . 3 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
3 | 1, 2 | mpbi 231 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴) |
4 | 3 | simpri 486 | 1 ⊢ ¬ 𝐵 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ⊆ wss 3933 ⊊ wpss 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ne 3014 df-in 3940 df-ss 3949 df-pss 3951 |
This theorem is referenced by: nsssmfmbf 42932 |
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