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Theorem oaord1 7198
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 4917 . . . 4  |-  (/)  e.  On
2 oaord 7194 . . . 4  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/) 
e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B ) ) )
31, 2mp3an1 1310 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  e.  B  <->  ( A  +o  (/) )  e.  ( A  +o  B
) ) )
4 oa0 7164 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
54adantl 466 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( A  +o  (/) )  =  A )
65eleq1d 2510 . . 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 1381    e. wcel 1802   (/)c0 3767   Oncon0 4864  (class class class)co 6277    +o coa 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-oadd 7132
This theorem is referenced by:  oaordex  7205  omordi  7213  wunex3  9117
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