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Theorem oaword2 6797
Description: An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. (Contributed by NM, 7-Dec-2004.)
Assertion
Ref Expression
oaword2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( B  +o  A ) )

Proof of Theorem oaword2
StepHypRef Expression
1 0ss 3657 . . 3  |-  (/)  C_  B
2 0elon 4635 . . . . 5  |-  (/)  e.  On
3 oawordri 6794 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  ( (/)  +o  A )  C_  ( B  +o  A ) ) )
42, 3mp3an1 1267 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  -> 
( (/)  +o  A ) 
C_  ( B  +o  A ) ) )
5 oa0r 6783 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
65adantl 454 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  +o  A
)  =  A )
76sseq1d 3376 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( (/)  +o  A
)  C_  ( B  +o  A )  <->  A  C_  ( B  +o  A ) ) )
84, 7sylibd 207 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  A  C_  ( B  +o  A ) ) )
91, 8mpi 17 . 2  |-  ( ( B  e.  On  /\  A  e.  On )  ->  A  C_  ( B  +o  A ) )
109ancoms 441 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( B  +o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   (/)c0 3629   Oncon0 4582  (class class class)co 6082    +o coa 6722
This theorem is referenced by:  oawordeulem  6798  nnarcl  6860  oaabslem  6887  oaabs2  6889  cantnfle  7627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-recs 6634  df-rdg 6669  df-oadd 6729
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