Theorem ordtr1 6202

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 6173	. 2 ⊢ (Ord 𝐶 → Tr 𝐶)
2		trel 5143	. 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 399   ∈ wcel 2111  Tr wtr 5136  Ord word 6158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137  df-ord 6162

This theorem is referenced by:  ontr1  6205  dfsmo2  7967  smores2  7974  smoel  7980  smogt  7987  ordiso2  8963  r1ordg  9191  r1pwss  9197  r1val1  9199  rankr1ai  9211  rankval3b  9239  rankonidlem  9241  onssr1  9244  cofsmo  9680  fpwwe2lem9  10049  bnj1098  32165  bnj594  32294  nosepssdm  33303