Theorem findset 9405

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.)

Assertion

Ref	Expression
findset	⊢ (A ∈ 𝑉 → ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → A = 𝜔))

Distinct variable group:   x,A

Allowed substitution hint:   𝑉(x)

Proof of Theorem findset

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))

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