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Theorem oaword2 6547
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 3485 . . 3  |-  (/)  C_  B
2 0elon 4445 . . . . 5  |-  (/)  e.  On
3 oawordri 6544 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  ( (/)  +o  A )  C_  ( B  +o  A ) ) )
42, 3mp3an1 1266 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  -> 
( (/)  +o  A ) 
C_  ( B  +o  A ) ) )
5 oa0r 6533 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
65adantl 454 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  +o  A
)  =  A )
76sseq1d 3207 . . . 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 18 . 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 6    /\ wa 360    = wceq 1624    e. wcel 1685    C_ wss 3154   (/)c0 3457   Oncon0 4392  (class class class)co 5820    +o coa 6472
This theorem is referenced by:  oawordeulem  6548  nnarcl  6610  oaabslem  6637  oaabs2  6639  cantnfle  7368
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-oadd 6479
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