Theorem ordelss 2215

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

Assertion

Ref	Expression
ordelss	⊢ ((Ord A ∧ B ∈ A) → B ⊆ A)

Proof of Theorem ordelss

Step	Hyp	Ref	Expression
1		trss 2050	. . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A))
2	1	imp 277	. 2 ⊢ ((Tr A ∧ B ∈ A) → B ⊆ A)
3		ordtr 2213	. 2 ⊢ (Ord A → Tr A)
4	2, 3	sylan 343	1 ⊢ ((Ord A ∧ B ∈ A) → B ⊆ A)

Syntax hints:   → wi 2   ∧ wa 196   ∈ 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

metamath.org