Theorem sselii 1505

Description: Membership inference from subclass relationship.

Hypotheses

Ref	Expression
sseli.1	⊢ A ⊆ B
sselii.2	⊢ C ∈ A

Assertion

Ref	Expression
sselii	⊢ C ∈ B

Proof of Theorem sselii

Step	Hyp	Ref	Expression
1		sselii.2	. 2 ⊢ C ∈ A
2		sseli.1	. . 3 ⊢ A ⊆ B
3	2	sseli 1504	. 2 ⊢ (C ∈ A → C ∈ B)
4	1, 3	ax-mp 6	1 ⊢ C ∈ B

Syntax hints:   ∈ wcel 1092   ⊆ wss 1487

This theorem is referenced by:  tz7.44-1 2966  tz7.44-2 2967  recn 4098  nn0re 4544  1nn0 4547  2nn0 4548  nthruc 4784  sheli 5121  cheli 5138  omlsilem 5249  pjssm 5572

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

metamath.org