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Mirrors > Home > ILE Home > Th. List > nzrring | GIF version |
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.) |
Ref | Expression |
---|---|
nzrring | ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nzr 13460 | . . 3 ⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)} | |
2 | 1 | ssrab3 3253 | . 2 ⊢ NzRing ⊆ Ring |
3 | 2 | sseli 3163 | 1 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 ≠ wne 2357 ‘cfv 5228 0gc0g 12723 1rcur 13268 Ringcrg 13305 NzRingcnzr 13459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-rab 2474 df-in 3147 df-ss 3154 df-nzr 13460 |
This theorem is referenced by: nzrunit 13465 lringring 13471 |
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