MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ontr1 Structured version   Visualization version   GIF version

Theorem ontr1 5759
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 5721 . 2 (𝐶 ∈ On → Ord 𝐶)
2 ordtr1 5755 . 2 (Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
31, 2syl 17 1 (𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  Ord word 5710  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-tr 4744  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715
This theorem is referenced by:  smoiun  7443  dif20el  7570  oeordi  7652  omabs  7712  omsmolem  7718  cofsmo  9076  cfsmolem  9077  inar1  9582  grur1a  9626  nosupno  31823  nosupbnd2lem1  31835
  Copyright terms: Public domain W3C validator