Theorem ordelss 2954

Description: An element of an ordinal class is a subset of it.

Assertion

Ref	Expression
ordelss	|- ((Ord A /\ B e. A) -> B (_ A)

Proof of Theorem ordelss

Step	Hyp	Ref	Expression
1		trss 2679	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2	1	imp 350	. 2 |- ((Tr A /\ B e. A) -> B (_ A)
3		ordtr 2952	. 2 |- (Ord A -> Tr A)
4	2, 3	sylan 448	1 |- ((Ord A /\ B e. A) -> B (_ A)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955   (_ wss 2037  Tr wtr 2670  Ord word 2937

This theorem is referenced by:  ordtri2or2 3068  oaabslem 4235  omsdomnn 4509

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-ral 1641  df-v 1803  df-in 2041  df-ss 2043  df-uni 2494  df-tr 2671  df-ord 2941

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