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Theorem oaord1 6982
Description: An ordinal is less than its sum with a nonzero ordinal. Theorem 18 of [Suppes] p. 209 and its converse. (Contributed by NM, 6-Dec-2004.)
Assertion
Ref Expression
oaord1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  A  e.  ( A  +o  B ) ) )

Proof of Theorem oaord1
StepHypRef Expression
1 0elon 4767 . . . 4  |-  (/)  e.  On
2 oaord 6978 . . . 4  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/) 
e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
31, 2mp3an1 1301 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
4 oa0 6948 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
54adantl 466 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( A  +o  (/) )  =  A )
65eleq1d 2504 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( A  +o  (/) )  e.  ( A  +o  B )  <->  A  e.  ( A  +o  B
) ) )
73, 6bitrd 253 . 2  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  B  <->  A  e.  ( A  +o  B ) ) )
87ancoms 453 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  e.  B  <->  A  e.  ( A  +o  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   (/)c0 3632   Oncon0 4714  (class class class)co 6086    +o coa 6909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-oadd 6916
This theorem is referenced by:  oaordex  6989  omordi  6997  wunex3  8900
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