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Mirrors > Home > MPE Home > Th. List > s1nz | Structured version Visualization version GIF version |
Description: A singleton word is not the empty string. (Contributed by Mario Carneiro, 27-Feb-2016.) (Proof shortened by Kyle Wyonch, 18-Jul-2021.) |
Ref | Expression |
---|---|
s1nz | ⊢ 〈“𝐴”〉 ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-s1 13944 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | opex 5348 | . . 3 ⊢ 〈0, ( I ‘𝐴)〉 ∈ V | |
3 | 2 | snnz 4704 | . 2 ⊢ {〈0, ( I ‘𝐴)〉} ≠ ∅ |
4 | 1, 3 | eqnetri 3086 | 1 ⊢ 〈“𝐴”〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 3016 ∅c0 4290 {csn 4560 〈cop 4566 I cid 5453 ‘cfv 6349 0cc0 10531 〈“cs1 13943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-s1 13944 |
This theorem is referenced by: lswccats1 13987 efgs1 18855 |
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