Theorem findset 7167

Description: Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4249 for a nonconstructive proof of the general case. See bdfind 7168 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 898	. . 3 ⊢ ((A ∈ 𝑉 ∧ (A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A)) → A ⊆ 𝜔)
2		simp2 893	. . . . . 6 ⊢ ((A ⊆ 𝜔 ∧ ∅ ∈ A ∧ ∀x ∈ A suc x ∈ A) → ∅ ∈ A)
3		df-ral 2289	. . . . . . . 8 ⊢ (∀x ∈ A suc x ∈ A ↔ ∀x(x ∈ A → suc x ∈ A))
4		alral 2345	. . . . . . . 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 915	. . . . . 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 877	. . . . . 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 7166	. . . 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 2939	. 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 873  ∀wal 1226   = wceq 1228   ∈ wcel 1374  ∀wral 2284   ⊆ wss 2894  ∅c0 3201  suc csuc 4051  𝜔com 4240

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdan 7042  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150

This theorem is referenced by:  bdfind  7168