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Theorem ontr1 5932
 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 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))

Proof of Theorem ontr1
StepHypRef Expression
1 eloni 5894 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 5928 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  Ord word 5883  Oncon0 5884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-tr 4905  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888 This theorem is referenced by:  smoiun  7627  dif20el  7754  oeordi  7836  omabs  7896  omsmolem  7902  cofsmo  9283  cfsmolem  9284  inar1  9789  grur1a  9833  nosupno  32155  nosupbnd2lem1  32167
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