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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 4265 for a nonconstructive proof of the general case. See bdfind 9406 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 909 | . . 3 ⊢ ((A ∈ 𝑉 ∧ (A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) → A ⊆ 𝜔) | |
2 | simp2 904 | . . . . . 6 ⊢ ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → ∅ ∈ A) | |
3 | df-ral 2305 | . . . . . . . 8 ⊢ (∀x ∈ A suc x ∈ A ↔ ∀x(x ∈ A → suc x ∈ A)) | |
4 | alral 2361 | . . . . . . . 8 ⊢ (∀x(x ∈ A → suc x ∈ A) → ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) | |
5 | 3, 4 | sylbi 114 | . . . . . . 7 ⊢ (∀x ∈ A suc x ∈ A → ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) |
6 | 5 | 3ad2ant3 926 | . . . . . 6 ⊢ ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) |
7 | 2, 6 | jca 290 | . . . . 5 ⊢ ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) |
8 | 3anass 888 | . . . . . 6 ⊢ ((A ∈ 𝑉 ∧ ∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) ↔ (A ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)))) | |
9 | 8 | biimpri 124 | . . . . 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 270 | . . . 4 ⊢ ((A ∈ 𝑉 ∧ (A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) → (A ∈ 𝑉 ∧ ∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) |
11 | speano5 9404 | . . . 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 2956 | . 2 ⊢ ((A ∈ 𝑉 ∧ (A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) → A = 𝜔) |
14 | 13 | ex 108 | 1 ⊢ (A ∈ 𝑉 → ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → A = 𝜔)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ⊆ wss 2911 ∅c0 3218 suc csuc 4068 𝜔com 4256 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-nul 3874 ax-pr 3935 ax-un 4136 ax-bd0 9268 ax-bdan 9270 ax-bdor 9271 ax-bdex 9274 ax-bdeq 9275 ax-bdel 9276 ax-bdsb 9277 ax-bdsep 9339 ax-infvn 9401 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 df-bdc 9296 df-bj-ind 9386 |
This theorem is referenced by: bdfind 9406 |
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