Theorem ordtr1 6438

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 6409	. 2 ⊢ (Ord 𝐶 → Tr 𝐶)
2		trel 5292	. 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 2108  Tr wtr 5283  Ord word 6394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-uni 4932  df-tr 5284  df-ord 6398

This theorem is referenced by:  ontr1  6441  dfsmo2  8403  smores2  8410  smoel  8416  smogt  8423  ordiso2  9584  r1ordg  9847  r1pwss  9853  r1val1  9855  rankr1ai  9867  rankval3b  9895  rankonidlem  9897  onssr1  9900  cofsmo  10338  fpwwe2lem8  10707  nosepssdm  27749  bnj1098  34759  bnj594  34888