Theorem bj-indsuc 10723

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.

Step	Hyp	Ref	Expression
1		df-bj-ind 10722	. . 3 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2	1	simprbi 269	. 2 ⊢ (Ind 𝐴 → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
3		suceq 4157	. . . 4 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2147	. . 3 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspcv 2697	. 2 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))
6	2, 5	syl5com 29	1 ⊢ (Ind 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  ∀wral 2348  ∅c0 3251  suc csuc 4120  Ind wind 10721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126  df-bj-ind 10722

This theorem is referenced by:  bj-indint  10726  bj-peano2  10734  bj-inf2vnlem2  10766