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Mirrors > Home > MPE Home > Th. List > Mathboxes > pssn0 | Structured version Visualization version GIF version |
Description: A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.) |
Ref | Expression |
---|---|
pssn0 | ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | npss0 4393 | . . 3 ⊢ ¬ 𝐴 ⊊ ∅ | |
2 | psseq2 4062 | . . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊊ 𝐵 ↔ 𝐴 ⊊ ∅)) | |
3 | 1, 2 | mtbiri 328 | . 2 ⊢ (𝐵 = ∅ → ¬ 𝐴 ⊊ 𝐵) |
4 | 3 | necon2ai 3042 | 1 ⊢ (𝐴 ⊊ 𝐵 → 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ≠ wne 3013 ⊊ wpss 3934 ∅c0 4288 |
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-dif 3936 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 |
This theorem is referenced by: xppss12 38993 |
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