Theorem sseldd 1507

Description: Membership inference from subclass relationship.

Hypotheses

Ref	Expression
sseld.1	⊢ (φ → A ⊆ B)
sseldd.2	⊢ (φ → C ∈ A)

Assertion

Ref	Expression
sseldd	⊢ (φ → C ∈ B)

Proof of Theorem sseldd

Step	Hyp	Ref	Expression
1		sseldd.2	. 2 ⊢ (φ → C ∈ A)
2		sseld.1	. . 3 ⊢ (φ → A ⊆ B)
3	2	sseld 1506	. 2 ⊢ (φ → (C ∈ A → C ∈ B))
4	1, 3	mpd 46	1 ⊢ (φ → C ∈ B)

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

This theorem is referenced by:  omordi 3164  tz9.12lem3 3505  pjhclt 5248  sumdmd 5787

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