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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 4508 for a nonconstructive proof of the general case. See findset 13132 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 13129 | . . . 4 ⊢ ω ∈ V | |
3 | 1, 2 | bdssex 13089 | . . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V) |
4 | 3 | 3ad2ant1 1002 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V) |
5 | findset 13132 | . 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 962 = wceq 1331 ∈ wcel 1480 ∀wral 2414 Vcvv 2681 ⊆ wss 3066 ∅c0 3358 suc csuc 4282 ωcom 4499 BOUNDED wbdc 13027 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-bd0 13000 ax-bdan 13002 ax-bdor 13003 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 ax-infvn 13128 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: (None) |
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