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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdpeano5 | GIF version |
Description: Bounded version of peano5 4612. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdpeano5.bd | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdpeano5 | ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdpeano5.bd | . . 3 ⊢ BOUNDED 𝐴 | |
2 | bj-omex 15132 | . . 3 ⊢ ω ∈ V | |
3 | 1, 2 | bdinex1 15089 | . 2 ⊢ (ω ∩ 𝐴) ∈ V |
4 | peano5set 15130 | . 2 ⊢ ((ω ∩ 𝐴) ∈ V → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 ∩ cin 3143 ⊆ wss 3144 ∅c0 3437 suc csuc 4380 ωcom 4604 BOUNDED wbdc 15030 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-nul 4144 ax-pr 4224 ax-un 4448 ax-bd0 15003 ax-bdor 15006 ax-bdex 15009 ax-bdeq 15010 ax-bdel 15011 ax-bdsb 15012 ax-bdsep 15074 ax-infvn 15131 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4386 df-iom 4605 df-bdc 15031 df-bj-ind 15117 |
This theorem is referenced by: bj-bdfindis 15137 |
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