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Mirrors > Home > ILE Home > Th. List > Mathboxes > peano5set | GIF version |
Description: Version of peano5 4580 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 13934 | . . . . 5 ⊢ Ind ω | |
2 | bj-indind 13932 | . . . . 5 ⊢ ((Ind ω ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → Ind (ω ∩ 𝐴)) | |
3 | 1, 2 | mpan 422 | . . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → Ind (ω ∩ 𝐴)) |
4 | bj-omssind 13935 | . . . . 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 3348 | . . 3 ⊢ (ω ∩ 𝐴) ⊆ 𝐴 | |
8 | 6, 7 | sstrdi 3159 | . 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 2141 ∀wral 2448 ∩ cin 3120 ⊆ wss 3121 ∅c0 3414 suc csuc 4348 ωcom 4572 Ind wind 13926 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4113 ax-pr 4192 ax-un 4416 ax-bd0 13813 ax-bdor 13816 ax-bdex 13819 ax-bdeq 13820 ax-bdel 13821 ax-bdsb 13822 ax-bdsep 13884 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3587 df-pr 3588 df-uni 3795 df-int 3830 df-suc 4354 df-iom 4573 df-bdc 13841 df-bj-ind 13927 |
This theorem is referenced by: bdpeano5 13943 speano5 13944 |
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