Theorem bj-indsuc 15864

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 15863	. . 3 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2	1	simprbi 275	. 2 ⊢ (Ind 𝐴 → ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)
3		suceq 4449	. . . 4 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
4	3	eleq1d 2274	. . 3 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
5	4	rspcv 2873	. 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 1373   ∈ wcel 2176  ∀wral 2484  ∅c0 3460  suc csuc 4412  Ind wind 15862

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-sn 3639  df-suc 4418  df-bj-ind 15863

This theorem is referenced by:  bj-indint  15867  bj-peano2  15875  bj-inf2vnlem2  15907