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Mirrors > Home > ILE Home > Th. List > Mathboxes > findset | GIF version |
Description: Bounded induction (principle of induction when 𝐴 is assumed to be a set) allowing a proof from basic constructive axioms. See find 4600 for a nonconstructive proof of the general case. See bdfind 14838 for a proof when 𝐴 is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
findset | ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1003 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) → 𝐴 ⊆ ω) | |
2 | simp2 998 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴) | |
3 | df-ral 2460 | . . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) | |
4 | alral 2522 | . . . . . . . 8 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | sylbi 121 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) |
6 | 5 | 3ad2ant3 1020 | . . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) |
7 | 2, 6 | jca 306 | . . . . 5 ⊢ ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) |
8 | 3anass 982 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ↔ (𝐴 ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))) | |
9 | 8 | biimpri 133 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) |
10 | 7, 9 | sylan2 286 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) |
11 | speano5 14836 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | |
12 | 10, 11 | syl 14 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) |
13 | 1, 12 | eqssd 3174 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) → 𝐴 = ω) |
14 | 13 | ex 115 | 1 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → 𝐴 = ω)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∀wal 1351 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3131 ∅c0 3424 suc csuc 4367 ωcom 4591 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4131 ax-pr 4211 ax-un 4435 ax-bd0 14705 ax-bdan 14707 ax-bdor 14708 ax-bdex 14711 ax-bdeq 14712 ax-bdel 14713 ax-bdsb 14714 ax-bdsep 14776 ax-infvn 14833 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-sn 3600 df-pr 3601 df-uni 3812 df-int 3847 df-suc 4373 df-iom 4592 df-bdc 14733 df-bj-ind 14819 |
This theorem is referenced by: bdfind 14838 |
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