Theorem sseldd 2058

Description: Membership inference from subclass relationship.

Hypotheses

Ref	Expression
sseld.1	|- (ph -> A (_ B)
sseldd.2	|- (ph -> C e. A)

Assertion

Ref	Expression
sseldd	|- (ph -> C e. B)

Proof of Theorem sseldd

Step	Hyp	Ref	Expression
1		sseldd.2	. 2 |- (ph -> C e. A)
2		sseld.1	. . 3 |- (ph -> A (_ B)
3	2	sseld 2057	. 2 |- (ph -> (C e. A -> C e. B))
4	1, 3	mpd 26	1 |- (ph -> C e. B)

Syntax hints:   -> wi 3   e. wcel 955   (_ wss 2037

This theorem is referenced by:  omordi 4181  tz9.12lem3 4633  fzelp1t 6440  seqzcl 6490  fsum0diag2 7194  acdc3lem 7428  acdc2lem1 7430  acdc5lem1 7433  acdclem 7436  iooretop 7601  metidcn 7839  bcthlem19 7951  bcthlem27 7959  subgid 8057  sspz 8328  pjhclt 9158  shatomistic 10196  sumdmdlem2 10253

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-in 2041  df-ss 2043

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