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Theorem peano5set 9399
Description: Version of peano5 4264 when 𝜔 ∩ 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
peano5set ((𝜔 ∩ 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 peano5set
StepHypRef Expression
1 bj-omind 9393 . . . . 5 Ind 𝜔
2 bj-indind 9391 . . . . 5 ((Ind 𝜔 (∅ A x 𝜔 (x A → suc x A))) → Ind (𝜔 ∩ A))
31, 2mpan 400 . . . 4 ((∅ A x 𝜔 (x A → suc x A)) → Ind (𝜔 ∩ A))
4 bj-omssind 9394 . . . . 5 ((𝜔 ∩ A) 𝑉 → (Ind (𝜔 ∩ A) → 𝜔 ⊆ (𝜔 ∩ A)))
54imp 115 . . . 4 (((𝜔 ∩ A) 𝑉 Ind (𝜔 ∩ A)) → 𝜔 ⊆ (𝜔 ∩ A))
63, 5sylan2 270 . . 3 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → 𝜔 ⊆ (𝜔 ∩ A))
7 inss2 3152 . . 3 (𝜔 ∩ A) ⊆ A
86, 7syl6ss 2951 . 2 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → 𝜔 ⊆ A)
98ex 108 1 ((𝜔 ∩ A) 𝑉 → ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  cin 2910  wss 2911  c0 3218  suc csuc 4068  𝜔com 4256  Ind wind 9385
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-bdor 9271  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339
This theorem depends on definitions:  df-bi 110  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:  bdpeano5  9403  speano5  9404
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