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Theorem oaword2 7239
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 3768 . . 3  |-  (/)  C_  B
2 0elon 5463 . . . . 5  |-  (/)  e.  On
3 oawordri 7236 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  ( (/)  +o  A )  C_  ( B  +o  A ) ) )
42, 3mp3an1 1313 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  -> 
( (/)  +o  A ) 
C_  ( B  +o  A ) ) )
5 oa0r 7225 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
65adantl 464 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  +o  A
)  =  A )
76sseq1d 3469 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( (/)  +o  A
)  C_  ( B  +o  A )  <->  A  C_  ( B  +o  A ) ) )
84, 7sylibd 214 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  A  C_  ( B  +o  A ) ) )
91, 8mpi 20 . 2  |-  ( ( B  e.  On  /\  A  e.  On )  ->  A  C_  ( B  +o  A ) )
109ancoms 451 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( B  +o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   (/)c0 3738   Oncon0 5410  (class class class)co 6278    +o coa 7164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-oadd 7171
This theorem is referenced by:  oawordeulem  7240  nnarcl  7302  oaabslem  7329  oaabs2  7331  cantnfle  8122  cantnfleOLD  8152
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