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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 13686 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | opex 5164 | . . 3 ⊢ 〈0, ( I ‘𝐴)〉 ∈ V | |
3 | 2 | snnz 4541 | . 2 ⊢ {〈0, ( I ‘𝐴)〉} ≠ ∅ |
4 | 1, 3 | eqnetri 3038 | 1 ⊢ 〈“𝐴”〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2968 ∅c0 4140 {csn 4397 〈cop 4403 I cid 5260 ‘cfv 6135 0cc0 10272 〈“cs1 13685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-s1 13686 |
This theorem is referenced by: lswccats1 13724 efgs1 18532 |
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