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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 4610 for a nonconstructive proof of the general case. See findset 14924 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 14921 | . . . 4 ⊢ ω ∈ V | |
3 | 1, 2 | bdssex 14881 | . . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ∈ V) |
4 | 3 | 3ad2ant1 1019 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 ∈ V) |
5 | findset 14924 | . 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 979 = wceq 1363 ∈ wcel 2158 ∀wral 2465 Vcvv 2749 ⊆ wss 3141 ∅c0 3434 suc csuc 4377 ωcom 4601 BOUNDED wbdc 14819 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-nul 4141 ax-pr 4221 ax-un 4445 ax-bd0 14792 ax-bdan 14794 ax-bdor 14795 ax-bdex 14798 ax-bdeq 14799 ax-bdel 14800 ax-bdsb 14801 ax-bdsep 14863 ax-infvn 14920 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-sn 3610 df-pr 3611 df-uni 3822 df-int 3857 df-suc 4383 df-iom 4602 df-bdc 14820 df-bj-ind 14906 |
This theorem is referenced by: (None) |
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