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Theorem findset 9333
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 9334 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 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 Ax(x A → suc x A))
4 alral 2361 . . . . . . . 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 926 . . . . . 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 888 . . . . . 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 9332 . . . 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 2956 . 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 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-bnd 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 9202  ax-bdan 9204  ax-bdor 9205  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273  ax-infvn 9329
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 9230  df-bj-ind 9316
This theorem is referenced by:  bdfind  9334
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