Theorem ontr2 5772

Description: Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)

Assertion

Ref	Expression
ontr2	|- ( ( A e. On /\ C e. On ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )

Proof of Theorem ontr2

Step	Hyp	Ref	Expression
1		eloni 5733	. 2 |- ( A e. On -> Ord A )
2		eloni 5733	. 2 |- ( C e. On -> Ord C )
3		ordtr2 5768	. 2 |- ( ( Ord A /\ Ord C ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )
4	1, 2, 3	syl2an 494	1 |- ( ( A e. On /\ C e. On ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  oeordsuc  7674  oelimcl  7680  oeeui  7682  omopthlem2  7736  omxpenlem  8061  oismo  8445  cantnflem1c  8584  cantnflem1  8586  cantnflem3  8588  rankr1ai  8661  rankxplim  8742  infxpenlem  8836  alephle  8911  pwcfsdom  9405  r1limwun  9558  ontopbas  32427  ontgval  32430