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Theorem speano5 7166
Description: Version of peano5 4248 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 7164 . . . 4 𝜔 V
2 bj-inex 7130 . . . 4 ((𝜔 V A 𝑉) → (𝜔 ∩ A) V)
31, 2mpan 402 . . 3 (A 𝑉 → (𝜔 ∩ A) V)
4 peano5set 7162 . . 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 1088 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 873   wcel 1374  wral 2284  Vcvv 2535  cin 2893  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:  findset  7167
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