Theorem sucidg 2305

Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized).

Assertion

Ref	Expression
sucidg	⊢ (A ∈ B → A ∈ suc A)

Proof of Theorem sucidg

Step	Hyp	Ref	Expression
1		id 9	. . 3 ⊢ (x = A → x = A)
2		suceq 2288	. . 3 ⊢ (x = A → suc x = suc A)
3	1, 2	eleq12d 1157	. 2 ⊢ (x = A → (x ∈ suc x ↔ A ∈ suc A))
4		visset 1350	. . 3 ⊢ x ∈ V
5	4	sucid 2304	. 2 ⊢ x ∈ suc x
6	3, 5	vtoclg 1383	1 ⊢ (A ∈ B → A ∈ suc A)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  suc csuc 2201

This theorem is referenced by:  nsuceq0 2306  trsuc 2308  ordsuc 2318  sucssel 2321  tfrlem13 2961  php4 3412  suc11reg 3456

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-suc 2205

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