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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-indsuc | GIF version |
Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.) |
Ref | Expression |
---|---|
bj-indsuc | ⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-ind 13125 | . . 3 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | 1 | simprbi 273 | . 2 ⊢ (Ind 𝐴 → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) |
3 | suceq 4324 | . . . 4 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
4 | 3 | eleq1d 2208 | . . 3 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
5 | 4 | rspcv 2785 | . 2 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
6 | 2, 5 | syl5com 29 | 1 ⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∅c0 3363 suc csuc 4287 Ind wind 13124 |
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-ral 2421 df-v 2688 df-un 3075 df-sn 3533 df-suc 4293 df-bj-ind 13125 |
This theorem is referenced by: bj-indint 13129 bj-peano2 13137 bj-inf2vnlem2 13169 |
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