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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 4295 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) | |
2 | ne0i 3921 | . 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 1990 ≠ wne 2794 ∅c0 3915 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-nul 3916 df-sn 4178 df-pr 4180 |
This theorem is referenced by: 0nelop 4960 fr2nr 5092 mreincl 16259 subrgin 18803 lssincl 18965 incld 20847 umgrnloopv 26001 upgr1elem 26007 usgrnloopvALT 26093 difelsiga 30196 inelpisys 30217 inidl 33829 pmapmeet 35059 diameetN 36345 dihmeetlem2N 36588 dihmeetcN 36591 dihmeet 36632 |
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