Theorem ontr1 6240

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 6204	. 2 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordtr1 6237	. 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 2113  Ord word 6193  Oncon0 6194

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-v 3499  df-in 3946  df-ss 3955  df-uni 4842  df-tr 5176  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-ord 6197  df-on 6198

This theorem is referenced by:  smoiun  8001  dif20el  8133  oeordi  8216  omabs  8277  omsmolem  8283  cofsmo  9694  cfsmolem  9695  inar1  10200  grur1a  10244  nosupno  33207  nosupbnd2lem1  33219