Theorem ordtr1 6406

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 6377	. 2 ⊢ (Ord 𝐶 → Tr 𝐶)
2		trel 5273	. 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 394   ∈ wcel 2104  Tr wtr 5264  Ord word 6362

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-tr 5265  df-ord 6366

This theorem is referenced by:  ontr1  6409  dfsmo2  8349  smores2  8356  smoel  8362  smogt  8369  ordiso2  9512  r1ordg  9775  r1pwss  9781  r1val1  9783  rankr1ai  9795  rankval3b  9823  rankonidlem  9825  onssr1  9828  cofsmo  10266  fpwwe2lem8  10635  nosepssdm  27425  bnj1098  34092  bnj594  34221