Theorem ordelss 2964

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 2689	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2	1	imp 350	. 2 |- ((Tr A /\ B e. A) -> B (_ A)
3		ordtr 2962	. 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 958   (_ wss 2047  Tr wtr 2680  Ord word 2947

This theorem is referenced by:  ordtri2or2 3078  oaabslem 4251  omsdomnn 4530

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-tr 2681  df-ord 2951

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