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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
StepHypRef 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 Ax(x A → suc x A))
4 alral 2345 . . . . . . . 8 (x(x A → suc x A) → x 𝜔 (x A → suc x A))
53, 4sylbi 114 . . . . . . 7 (x A suc x Ax 𝜔 (x A → suc x A))
653ad2ant3 915 . . . . . 6 ((A ⊆ 𝜔 A x A suc x A) → x 𝜔 (x A → suc x A))
72, 6jca 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))))
98biimpri 124 . . . . 5 ((A 𝑉 (∅ A x 𝜔 (x A → suc x A))) → (A 𝑉 A x 𝜔 (x A → suc x A)))
107, 9sylan2 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)
1210, 11syl 14 . . 3 ((A 𝑉 (A ⊆ 𝜔 A x A suc x A)) → 𝜔 ⊆ A)
131, 12eqssd 2939 . 2 ((A 𝑉 (A ⊆ 𝜔 A x A suc x A)) → A = 𝜔)
1413ex 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 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
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