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| 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 4760 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
| 2 | 1 | ne0d 4342 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {cpr 4628 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: preqsnd 4859 0nelop 5501 fr2nr 5662 mreincl 17642 subrngin 20561 subrgin 20596 lssincl 20963 incld 23051 umgrnloopv 29123 upgr1elem 29129 usgrnloopvALT 29218 difelsiga 34134 inelpisys 34155 inidl 38037 coss0 38480 pmapmeet 39775 diameetN 41058 dihmeetlem2N 41301 dihmeetcN 41304 dihmeet 41345 | 
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