Theorem sucssel 2321

Description: A set whose successor is a subset of another class is a member of that class.

Assertion

Ref	Expression
sucssel	⊢ (A ∈ C → (suc A ⊆ B → A ∈ B))

Proof of Theorem sucssel

Step	Hyp	Ref	Expression
1		ssel 1502	. . 3 ⊢ (suc A ⊆ B → (A ∈ suc A → A ∈ B))
2		sucidg 2305	. . 3 ⊢ (A ∈ C → A ∈ suc A)
3	1, 2	syl5 22	. 2 ⊢ (suc A ⊆ B → (A ∈ C → A ∈ B))
4	3	com12 13	1 ⊢ (A ∈ C → (suc A ⊆ B → A ∈ B))

Syntax hints:   → wi 2   ∈ wcel 1092   ⊆ wss 1487  suc csuc 2201

This theorem is referenced by:  ordelsuc 2322  ordsucelsuc 2324  suc11 2341  oaordi 3148  unbnn2 3436  r1ord 3499  cflim 3704  indpi 3828

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-in 1491  df-ss 1492  df-sn 1811  df-pr 1812  df-suc 2205

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