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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 701 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | elsucg 4326 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | mpbird 166 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 suc csuc 4287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-suc 4293 |
This theorem is referenced by: suctr 4343 ordsuc 4478 nnaordex 6423 fiintim 6817 exmidfodomrlemr 7058 exmidfodomrlemrALT 7059 ennnfonelemex 11927 |
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