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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 712 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | elsucg 4400 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | mpbird 167 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 suc csuc 4361 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-suc 4367 |
This theorem is referenced by: suctr 4417 ordsuc 4558 nnaordex 6522 fiintim 6921 exmidfodomrlemr 7194 exmidfodomrlemrALT 7195 3nelsucpw1 7226 ennnfonelemex 12385 |
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