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Theorem bj-indsuc 13140
Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indsuc (Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))

Proof of Theorem bj-indsuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 13139 . . 3 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
21simprbi 273 . 2 (Ind 𝐴 → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 4324 . . . 4 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2208 . . 3 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspcv 2785 . 2 (𝐵𝐴 → (∀𝑥𝐴 suc 𝑥𝐴 → suc 𝐵𝐴))
62, 5syl5com 29 1 (Ind 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  wral 2416  c0 3363  suc csuc 4287  Ind wind 13138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293  df-bj-ind 13139
This theorem is referenced by:  bj-indint  13143  bj-peano2  13151  bj-inf2vnlem2  13183
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