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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 4239 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | ne0i 3880 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵} → {𝐴, 𝐵} ≠ ∅) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐵} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: 0nelop 4886 fr2nr 5016 mreincl 16082 subrgin 18626 lssincl 18786 incld 20657 umgrnloopv 25772 upgr1elem 25778 umgra1 25855 uslgra1 25901 usgranloopv 25907 difelsiga 29523 inelpisys 29544 inidl 32999 pmapmeet 34077 diameetN 35363 dihmeetlem2N 35606 dihmeetcN 35609 dihmeet 35650 usgrnloopvALT 40428 |
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