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Theorem oaword1 7193
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 7192.) (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 7158 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
21adantr 463 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  =  A )
3 0ss 3813 . . 3  |-  (/)  C_  B
4 0elon 4920 . . . 4  |-  (/)  e.  On
5 oaword 7190 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
653com13 1199 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  (/)  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B
) ) )
74, 6mp3an3 1311 . . 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 3524 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 367    = wceq 1398    e. wcel 1823    C_ wss 3461   (/)c0 3783   Oncon0 4867  (class class class)co 6270    +o coa 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-oadd 7126
This theorem is referenced by:  oawordexr  7197  oa00  7200  oaf1o  7204  omordi  7207  omeulem2  7224  oeeui  7243  nnarcl  7257  omxpenlem  7611  cantnfle  8081  cantnflem1d  8098  cantnflem3  8101  cantnflem4  8102  cantnfleOLD  8111  cantnflem1dOLD  8121  cantnflem3OLD  8123  cantnflem4OLD  8124
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