Theorem ordtr1 6361

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 6331	. 2 ⊢ (Ord 𝐶 → Tr 𝐶)
2		trel 5213	. 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 2113  Tr wtr 5205  Ord word 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-ss 3918  df-uni 4864  df-tr 5206  df-ord 6320

This theorem is referenced by:  ontr1  6364  dfsmo2  8279  smores2  8286  smoel  8292  smogt  8299  ordiso2  9420  r1ordg  9690  r1pwss  9696  r1val1  9698  rankr1ai  9710  rankval3b  9738  rankonidlem  9740  onssr1  9743  cofsmo  10179  fpwwe2lem8  10549  nosepssdm  27654  bnj1098  34939  bnj594  35068  rankfilimb  35258  r1filimi  35259