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Mirrors > Home > ILE Home > Th. List > Mathboxes > peano5set | GIF version |
Description: Version of peano5 4520 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 13303 | . . . . 5 ⊢ Ind ω | |
2 | bj-indind 13301 | . . . . 5 ⊢ ((Ind ω ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → Ind (ω ∩ 𝐴)) | |
3 | 1, 2 | mpan 421 | . . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → Ind (ω ∩ 𝐴)) |
4 | bj-omssind 13304 | . . . . 5 ⊢ ((ω ∩ 𝐴) ∈ 𝑉 → (Ind (ω ∩ 𝐴) → ω ⊆ (ω ∩ 𝐴))) | |
5 | 4 | imp 123 | . . . 4 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ Ind (ω ∩ 𝐴)) → ω ⊆ (ω ∩ 𝐴)) |
6 | 3, 5 | sylan2 284 | . . 3 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → ω ⊆ (ω ∩ 𝐴)) |
7 | inss2 3302 | . . 3 ⊢ (ω ∩ 𝐴) ⊆ 𝐴 | |
8 | 6, 7 | sstrdi 3114 | . 2 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → ω ⊆ 𝐴) |
9 | 8 | ex 114 | 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 103 ∈ wcel 1481 ∀wral 2417 ∩ cin 3075 ⊆ wss 3076 ∅c0 3368 suc csuc 4295 ωcom 4512 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-in1 604 ax-in2 605 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-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-bd0 13182 ax-bdor 13185 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 |
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-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: bdpeano5 13312 speano5 13313 |
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