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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 14634 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
| 2 | opex 5469 | . . 3 ⊢ 〈0, ( I ‘𝐴)〉 ∈ V | |
| 3 | 2 | snnz 4776 | . 2 ⊢ {〈0, ( I ‘𝐴)〉} ≠ ∅ |
| 4 | 1, 3 | eqnetri 3011 | 1 ⊢ 〈“𝐴”〉 ≠ ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ≠ wne 2940 ∅c0 4333 {csn 4626 〈cop 4632 I cid 5577 ‘cfv 6561 0cc0 11155 〈“cs1 14633 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 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-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-s1 14634 |
| This theorem is referenced by: lswccats1 14672 efgs1 19753 singoutnword 46897 |
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