Theorem ordtr1 2256

Description: Transitive law for ordinal classes.

Assertion

Ref	Expression
ordtr1	⊢ (Ord C → ((A ∈ B ∧ B ∈ C) → A ∈ C))

Proof of Theorem ordtr1

Step	Hyp	Ref	Expression
1		ordtr 2213	. 2 ⊢ (Ord C → Tr C)
2		trel 2048	. 2 ⊢ (Tr C → ((A ∈ B ∧ B ∈ C) → A ∈ C))
3	1, 2	syl 12	1 ⊢ (Ord C → ((A ∈ B ∧ B ∈ C) → A ∈ C))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Tr wtr 2041  Ord word 2198

This theorem is referenced by:  ordtr2 2257  ontr1 2258

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

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