Theorem ontr1 3003

Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192.

Assertion

Ref	Expression
ontr1	|- (C e. On -> ((A e. B /\ B e. C) -> A e. C))

Proof of Theorem ontr1

Step	Hyp	Ref	Expression
1		eloni 2958	. 2 |- (C e. On -> Ord C)
2		ordtr1 3001	. 2 |- (Ord C -> ((A e. B /\ B e. C) -> A e. C))
3	1, 2	syl 10	1 |- (C e. On -> ((A e. B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Ord word 2947  Oncon0 2948

This theorem is referenced by:  oeordi 4214  omsmolem 4256  ltsopi 5016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-tr 2681  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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