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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 9386 | . . 3 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) | |
2 | 1 | simprbi 260 | . 2 ⊢ (Ind A → ∀x ∈ A suc x ∈ A) |
3 | suceq 4105 | . . . 4 ⊢ (x = B → suc x = suc B) | |
4 | 3 | eleq1d 2103 | . . 3 ⊢ (x = B → (suc x ∈ A ↔ suc B ∈ A)) |
5 | 4 | rspcv 2646 | . 2 ⊢ (B ∈ A → (∀x ∈ A suc x ∈ A → suc B ∈ A)) |
6 | 2, 5 | syl5com 26 | 1 ⊢ (Ind A → (B ∈ A → suc B ∈ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ∅c0 3218 suc csuc 4068 Ind wind 9385 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-un 2916 df-sn 3373 df-suc 4074 df-bj-ind 9386 |
This theorem is referenced by: bj-indint 9390 bj-peano2 9398 bj-inf2vnlem2 9431 |
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