Theorem ontr1 6239

Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Assertion

Ref	Expression
ontr1	⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Proof of Theorem ontr1

Step	Hyp	Ref	Expression
1		eloni 6203	. 2 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordtr1 6236	. 2 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  Ord word 6192  Oncon0 6193

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-v 3498  df-in 3945  df-ss 3954  df-uni 4841  df-tr 5175  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197

This theorem is referenced by:  smoiun  8000  dif20el  8132  oeordi  8215  omabs  8276  omsmolem  8282  cofsmo  9693  cfsmolem  9694  inar1  10199  grur1a  10243  nosupno  33205  nosupbnd2lem1  33217