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Mirrors > Home > ILE Home > Th. List > Mathboxes > speano5 | GIF version |
Description: Version of peano5 4594 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 14350 | . . . 4 ⊢ ω ∈ V | |
2 | bj-inex 14315 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝑉) → (ω ∩ 𝐴) ∈ V) | |
3 | 1, 2 | mpan 424 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (ω ∩ 𝐴) ∈ V) |
4 | peano5set 14348 | . . 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 1201 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 ∅c0 3422 suc csuc 4362 ωcom 4586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4126 ax-pr 4206 ax-un 4430 ax-bd0 14221 ax-bdan 14223 ax-bdor 14224 ax-bdex 14227 ax-bdeq 14228 ax-bdel 14229 ax-bdsb 14230 ax-bdsep 14292 ax-infvn 14349 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3597 df-pr 3598 df-uni 3808 df-int 3843 df-suc 4368 df-iom 4587 df-bdc 14249 df-bj-ind 14335 |
This theorem is referenced by: findset 14353 |
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