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Theorem speano5 9332
Description: Version of peano5 4264 when A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
speano5 ((A 𝑉 A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Distinct variable group:   x,A
Allowed substitution hint:   𝑉(x)

Proof of Theorem speano5
StepHypRef Expression
1 bj-omex 9330 . . . 4 𝜔 V
2 bj-inex 9292 . . . 4 ((𝜔 V A 𝑉) → (𝜔 ∩ A) V)
31, 2mpan 400 . . 3 (A 𝑉 → (𝜔 ∩ A) V)
4 peano5set 9328 . . 3 ((𝜔 ∩ A) V → ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A))
53, 4syl 14 . 2 (A 𝑉 → ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A))
653impib 1101 1 ((A 𝑉 A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   wcel 1390  wral 2300  Vcvv 2551  cin 2910  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:  findset  9333
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