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Mirrors > Home > ILE Home > Th. List > sucssel | GIF version |
Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.) |
Ref | Expression |
---|---|
sucssel | ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sucidg 4333 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
2 | ssel 3086 | . 2 ⊢ (suc 𝐴 ⊆ 𝐵 → (𝐴 ∈ suc 𝐴 → 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | syl5com 29 | 1 ⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 ⊆ 𝐵 → 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ⊆ wss 3066 suc csuc 4282 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-suc 4288 |
This theorem is referenced by: ordelsuc 4416 bj-nnelirr 13140 |
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