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Theorem bj-indsuc 7151
Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indsuc (Ind A → (B A → suc B A))

Proof of Theorem bj-indsuc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 7150 . . 3 (Ind A ↔ (∅ A x A suc x A))
21simprbi 260 . 2 (Ind Ax A suc x A)
3 suceq 4088 . . . 4 (x = B → suc x = suc B)
43eleq1d 2088 . . 3 (x = B → (suc x A ↔ suc B A))
54rspcv 2629 . 2 (B A → (x A suc x A → suc B A))
62, 5syl5com 26 1 (Ind A → (B A → suc B A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  wral 2284  c0 3201  suc csuc 4051  Ind wind 7149
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-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-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
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-v 2537  df-un 2899  df-sn 3356  df-suc 4057  df-bj-ind 7150
This theorem is referenced by:  bj-indint  7154  bj-peano2  7161  bj-inf2vnlem2  7189
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