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Theorem oaword2 6519
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 3458 . . 3  |-  (/)  C_  B
2 0elon 4417 . . . . 5  |-  (/)  e.  On
3 oawordri 6516 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  ( (/)  +o  A )  C_  ( B  +o  A ) ) )
42, 3mp3an1 1269 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  -> 
( (/)  +o  A ) 
C_  ( B  +o  A ) ) )
5 oa0r 6505 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
65adantl 454 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  +o  A
)  =  A )
76sseq1d 3180 . . . 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 1619    e. wcel 1621    C_ wss 3127   (/)c0 3430   Oncon0 4364  (class class class)co 5792    +o coa 6444
This theorem is referenced by:  oawordeulem  6520  nnarcl  6582  oaabslem  6609  oaabs2  6611  cantnfle  7340
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-oadd 6451
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