Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13157 | . 2 ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} | |
2 | opex 4859 | . . 3 ⊢ 〈0, ( I ‘𝐴)〉 ∈ V | |
3 | 2 | snnz 4252 | . 2 ⊢ {〈0, ( I ‘𝐴)〉} ≠ ∅ |
4 | 1, 3 | eqnetri 2852 | 1 ⊢ 〈“𝐴”〉 ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2780 ∅c0 3874 {csn 4125 〈cop 4131 I cid 4948 ‘cfv 5804 0cc0 9815 〈“cs1 13149 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 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-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-s1 13157 |
This theorem is referenced by: lswccats1 13263 efgs1 17971 |
Copyright terms: Public domain | W3C validator |