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Mirrors > Home > ILE Home > Th. List > Mathboxes > speano5 | GIF version |
Description: Version of peano5 4569 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 13665 | . . . 4 ⊢ ω ∈ V | |
2 | bj-inex 13630 | . . . 4 ⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝑉) → (ω ∩ 𝐴) ∈ V) | |
3 | 1, 2 | mpan 421 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (ω ∩ 𝐴) ∈ V) |
4 | peano5set 13663 | . . 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 1190 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 ∈ wcel 2135 ∀wral 2442 Vcvv 2721 ∩ cin 3110 ⊆ wss 3111 ∅c0 3404 suc csuc 4337 ωcom 4561 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-bd0 13536 ax-bdan 13538 ax-bdor 13539 ax-bdex 13542 ax-bdeq 13543 ax-bdel 13544 ax-bdsb 13545 ax-bdsep 13607 ax-infvn 13664 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 df-bdc 13564 df-bj-ind 13650 |
This theorem is referenced by: findset 13668 |
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