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Theorem bj-indsuc 10439
 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 10438 . . 3 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
21simprbi 264 . 2 (Ind 𝐴 → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 4167 . . . 4 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2122 . . 3 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspcv 2669 . 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 1259   ∈ wcel 1409  ∀wral 2323  ∅c0 3252  suc csuc 4130  Ind wind 10437 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-suc 4136  df-bj-ind 10438 This theorem is referenced by:  bj-indint  10442  bj-peano2  10450  bj-inf2vnlem2  10483
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