Theorem bj-indsuc 15420

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 15419	. . 3 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2	1	simprbi 275	. 2 ⊢ (Ind 𝐴 → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
3		suceq 4433	. . . 4 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2262	. . 3 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspcv 2860	. 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 1364   ∈ wcel 2164  ∀wral 2472  ∅c0 3446  suc csuc 4396  Ind wind 15418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3157  df-sn 3624  df-suc 4402  df-bj-ind 15419

This theorem is referenced by:  bj-indint  15423  bj-peano2  15431  bj-inf2vnlem2  15463