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Mirrors > Home > ILE Home > Th. List > Mathboxes > speano5 | GIF version |
Description: Version of peano5 4413 when 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 |
---|---|
speano5 | ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-omex 11837 | . . . 4 ⊢ ω ∈ V | |
2 | bj-inex 11798 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝑉) → (ω ∩ 𝐴) ∈ V) | |
3 | 1, 2 | mpan 415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (ω ∩ 𝐴) ∈ V) |
4 | peano5set 11835 | . . 3 ⊢ ((ω ∩ 𝐴) ∈ V → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)) |
6 | 5 | 3impib 1141 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 924 ∈ wcel 1438 ∀wral 2359 Vcvv 2619 ∩ cin 2998 ⊆ wss 2999 ∅c0 3286 suc csuc 4192 ωcom 4405 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-nul 3965 ax-pr 4036 ax-un 4260 ax-bd0 11704 ax-bdan 11706 ax-bdor 11707 ax-bdex 11710 ax-bdeq 11711 ax-bdel 11712 ax-bdsb 11713 ax-bdsep 11775 ax-infvn 11836 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-suc 4198 df-iom 4406 df-bdc 11732 df-bj-ind 11822 |
This theorem is referenced by: findset 11840 |
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