Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > wrdexi | Structured version Visualization version GIF version |
Description: The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.) |
Ref | Expression |
---|---|
wrdexi.1 | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
wrdexi | ⊢ Word 𝑆 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdexi.1 | . 2 ⊢ 𝑆 ∈ V | |
2 | wrdexg 13966 | . 2 ⊢ (𝑆 ∈ V → Word 𝑆 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ Word 𝑆 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3398 Word cword 13956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-1cn 10674 ax-addcl 10676 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-map 8440 df-nn 11718 df-n0 11978 df-word 13957 |
This theorem is referenced by: ablfaclem1 19327 israg 26643 rusgrnumwrdl2 27528 ewlksfval 27543 mrexval 33034 |
Copyright terms: Public domain | W3C validator |