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Theorem oaword1 6547
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 6546.) (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 6512 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
21adantr 453 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  =  A )
3 0ss 3486 . . 3  |-  (/)  C_  B
4 0elon 4446 . . . 4  |-  (/)  e.  On
5 oaword 6544 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
653com13 1158 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  (/)  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B
) ) )
74, 6mp3an3 1268 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
83, 7mpbii 204 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  C_  ( A  +o  B
) )
92, 8eqsstr3d 3216 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687    C_ wss 3155   (/)c0 3458   Oncon0 4393  (class class class)co 5821    +o coa 6473
This theorem is referenced by:  oawordexr  6551  oa00  6554  oaf1o  6558  omordi  6561  omeulem2  6578  oeeui  6597  nnarcl  6611  omxpenlem  6960  cantnfle  7369  cantnflem1d  7387  cantnflem3  7390  cantnflem4  7391
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pr 4215  ax-un 4513
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 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-recs 6385  df-rdg 6420  df-oadd 6480
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