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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 702 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | elsucg 4376 | . 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 698 = wceq 1342 ∈ wcel 2135 suc csuc 4337 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-suc 4343 |
This theorem is referenced by: suctr 4393 ordsuc 4534 nnaordex 6486 fiintim 6885 exmidfodomrlemr 7149 exmidfodomrlemrALT 7150 3nelsucpw1 7181 ennnfonelemex 12290 |
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