Theorem ordelss 2215

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 2050	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2	1	imp 277	. 2 |- ((Tr A /\ B e. A) -> B (_ A)
3		ordtr 2213	. 2 |- (Ord A -> Tr A)
4	2, 3	sylan 343	1 |- ((Ord A /\ B e. A) -> B (_ A)

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092   (_ wss 1487  Tr wtr 2041  Ord word 2198

This theorem is referenced by:  ordtri2or2 2329  omsdomnn 3424

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-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-uni 1920  df-tr 2042  df-ord 2202

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