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Mirrors > Home > ILE Home > Th. List > elelsuc | GIF version |
Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.) |
Ref | Expression |
---|---|
elelsuc | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 665 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | elsucg 4159 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | mpbird 165 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 661 = wceq 1284 ∈ wcel 1433 suc csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-suc 4126 |
This theorem is referenced by: suctr 4176 ordsuc 4306 nnaordex 6123 |
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