Theorem bj-indsuc 9387

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 9386	. . 3 ⊢ (Ind A ↔ (∅ ∈ A ∧ ∀x ∈ A suc x ∈ A))
2	1	simprbi 260	. 2 ⊢ (Ind A → ∀x ∈ A suc x ∈ A)
3		suceq 4105	. . . 4 ⊢ (x = B → suc x = suc B)
4	3	eleq1d 2103	. . 3 ⊢ (x = B → (suc x ∈ A ↔ suc B ∈ A))
5	4	rspcv 2646	. 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 1242   ∈ wcel 1390  ∀wral 2300  ∅c0 3218  suc csuc 4068  Ind wind 9385

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-suc 4074  df-bj-ind 9386

This theorem is referenced by:  bj-indint  9390  bj-peano2  9398  bj-inf2vnlem2  9431