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Theorem bdpeano5 7312
Description: Bounded version of peano5 4248. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdpeano5.bd BOUNDED A
Assertion
Ref Expression
bdpeano5 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Distinct variable group:   x,A

Proof of Theorem bdpeano5
StepHypRef Expression
1 bdpeano5.bd . . 3 BOUNDED A
2 bj-omex 7311 . . 3 𝜔 V
31, 2bdinex1 7269 . 2 (𝜔 ∩ A) V
4 peano5set 7309 . 2 ((𝜔 ∩ A) V → ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A))
53, 4ax-mp 7 1 ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  wral 2284  Vcvv 2535  cin 2893  wss 2894  c0 3201  suc csuc 4051  𝜔com 4240  BOUNDED wbdc 7214
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 7187  ax-bdor 7190  ax-bdex 7193  ax-bdeq 7194  ax-bdel 7195  ax-bdsb 7196  ax-bdsep 7258  ax-infvn 7310
This theorem depends on definitions:  df-bi 110  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 7215  df-bj-ind 7297
This theorem is referenced by:  bj-bdfindis  7316
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