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Theorem oaword1 6991
Description: An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (For the other part see oaord1 6990.) (Contributed by NM, 6-Dec-2004.)
Assertion
Ref Expression
oaword1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )

Proof of Theorem oaword1
StepHypRef Expression
1 oa0 6956 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
21adantr 465 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  =  A )
3 0ss 3666 . . 3  |-  (/)  C_  B
4 0elon 4772 . . . 4  |-  (/)  e.  On
5 oaword 6988 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
653com13 1192 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  (/)  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B
) ) )
74, 6mp3an3 1303 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
83, 7mpbii 211 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  C_  ( A  +o  B
) )
92, 8eqsstr3d 3391 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   (/)c0 3637   Oncon0 4719  (class class class)co 6091    +o coa 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-oadd 6924
This theorem is referenced by:  oawordexr  6995  oa00  6998  oaf1o  7002  omordi  7005  omeulem2  7022  oeeui  7041  nnarcl  7055  omxpenlem  7412  cantnfle  7879  cantnflem1d  7896  cantnflem3  7899  cantnflem4  7900  cantnfleOLD  7909  cantnflem1dOLD  7919  cantnflem3OLD  7921  cantnflem4OLD  7922
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