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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 13296 | . . 3 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | 1 | simprbi 273 | . 2 ⊢ (Ind 𝐴 → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) |
3 | suceq 4332 | . . . 4 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
4 | 3 | eleq1d 2209 | . . 3 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
5 | 4 | rspcv 2789 | . 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 1332 ∈ wcel 1481 ∀wral 2417 ∅c0 3368 suc csuc 4295 Ind wind 13295 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-sn 3538 df-suc 4301 df-bj-ind 13296 |
This theorem is referenced by: bj-indint 13300 bj-peano2 13308 bj-inf2vnlem2 13340 |
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