MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oaword2 Unicode version

Theorem oaword2 6551
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 3483 . . 3  |-  (/)  C_  B
2 0elon 4445 . . . . 5  |-  (/)  e.  On
3 oawordri 6548 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  ( (/)  +o  A )  C_  ( B  +o  A ) ) )
42, 3mp3an1 1264 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  -> 
( (/)  +o  A ) 
C_  ( B  +o  A ) ) )
5 oa0r 6537 . . . . . 6  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
65adantl 452 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  +o  A
)  =  A )
76sseq1d 3205 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( ( (/)  +o  A
)  C_  ( B  +o  A )  <->  A  C_  ( B  +o  A ) ) )
84, 7sylibd 205 . . 3  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  ->  A  C_  ( B  +o  A ) ) )
91, 8mpi 16 . 2  |-  ( ( B  e.  On  /\  A  e.  On )  ->  A  C_  ( B  +o  A ) )
109ancoms 439 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( B  +o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   (/)c0 3455   Oncon0 4392  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  oawordeulem  6552  nnarcl  6614  oaabslem  6641  oaabs2  6643  cantnfle  7372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
  Copyright terms: Public domain W3C validator