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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdfind | GIF version |
Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4613 for a nonconstructive proof of the general case. See findset 15094 for a proof when 𝐴 is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdfind.bd | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdfind | ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdfind.bd | . . . 4 ⊢ BOUNDED 𝐴 | |
2 | bj-omex 15091 | . . . 4 ⊢ ω ∈ V | |
3 | 1, 2 | bdssex 15051 | . . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V) |
4 | 3 | 3ad2ant1 1020 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V) |
5 | findset 15094 | . 2 ⊢ (𝐴 ∈ V → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)) | |
6 | 4, 5 | mpcom 36 | 1 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 ⊆ wss 3144 ∅c0 3437 suc csuc 4380 ωcom 4604 BOUNDED wbdc 14989 |
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 14962 ax-bdan 14964 ax-bdor 14965 ax-bdex 14968 ax-bdeq 14969 ax-bdel 14970 ax-bdsb 14971 ax-bdsep 15033 ax-infvn 15090 |
This theorem depends on definitions: df-bi 117 df-3an 982 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 14990 df-bj-ind 15076 |
This theorem is referenced by: (None) |
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