Theorem ordtr1 6379

Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)

Assertion

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

Proof of Theorem ordtr1

Step	Hyp	Ref	Expression
1		ordtr 6349	. 2 ⊢ (Ord 𝐶 → Tr 𝐶)
2		trel 5226	. 2 ⊢ (Tr 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
3	1, 2	syl 17	1 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Tr wtr 5217  Ord word 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-uni 4875  df-tr 5218  df-ord 6338

This theorem is referenced by:  ontr1  6382  dfsmo2  8319  smores2  8326  smoel  8332  smogt  8339  ordiso2  9475  r1ordg  9738  r1pwss  9744  r1val1  9746  rankr1ai  9758  rankval3b  9786  rankonidlem  9788  onssr1  9791  cofsmo  10229  fpwwe2lem8  10598  nosepssdm  27605  bnj1098  34780  bnj594  34909