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Mirrors > Home > MPE Home > Th. List > prnzg | Structured version Visualization version GIF version |
Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
prnzg | ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 4688 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | 1 | ne0d 4298 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 ∅c0 4288 {cpr 4559 |
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-nfc 2960 df-ne 3014 df-v 3494 df-dif 3936 df-un 3938 df-nul 4289 df-sn 4558 df-pr 4560 |
This theorem is referenced by: preqsnd 4781 0nelop 5377 fr2nr 5526 mreincl 16858 subrgin 19487 lssincl 19666 incld 21579 umgrnloopv 26818 upgr1elem 26824 usgrnloopvALT 26910 difelsiga 31291 inelpisys 31312 inidl 35189 coss0 35599 pmapmeet 36789 diameetN 38072 dihmeetlem2N 38315 dihmeetcN 38318 dihmeet 38359 |
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