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Mirrors > Home > ILE Home > Th. List > Mathboxes > peano5set | GIF version |
Description: Version of peano5 4264 when 𝜔 ∩ A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
peano5set | ⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omind 9393 | . . . . 5 ⊢ Ind 𝜔 | |
2 | bj-indind 9391 | . . . . 5 ⊢ ((Ind 𝜔 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → Ind (𝜔 ∩ A)) | |
3 | 1, 2 | mpan 400 | . . . 4 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → Ind (𝜔 ∩ A)) |
4 | bj-omssind 9394 | . . . . 5 ⊢ ((𝜔 ∩ A) ∈ 𝑉 → (Ind (𝜔 ∩ A) → 𝜔 ⊆ (𝜔 ∩ A))) | |
5 | 4 | imp 115 | . . . 4 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ Ind (𝜔 ∩ A)) → 𝜔 ⊆ (𝜔 ∩ A)) |
6 | 3, 5 | sylan2 270 | . . 3 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → 𝜔 ⊆ (𝜔 ∩ A)) |
7 | inss2 3152 | . . 3 ⊢ (𝜔 ∩ A) ⊆ A | |
8 | 6, 7 | syl6ss 2951 | . 2 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → 𝜔 ⊆ A) |
9 | 8 | ex 108 | 1 ⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ∀wral 2300 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 suc csuc 4068 𝜔com 4256 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-in1 544 ax-in2 545 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-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdor 9271 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 |
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-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bdpeano5 9403 speano5 9404 |
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