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.

Step	Hyp	Ref	Expression
1		df-bj-ind 7150	. . 3 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A))
2	1	simprbi 260	. 2 ⊢ (Ind A → ∀x ∈ A suc x ∈ A)
3		suceq 4088	. . . 4 ⊢ (x = B → suc x = suc B)
4	3	eleq1d 2088	. . 3 ⊢ (x = B → (suc x ∈ A ↔ suc B ∈ A))
5	4	rspcv 2629	. 2 ⊢ (B ∈ A → (∀x ∈ A suc x ∈ A → suc B ∈ A))
6	2, 5	syl5com 26	1 ⊢ (Ind A → (B ∈ A → suc B ∈ A))

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