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Theorem oaword1 6592
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 6591.) (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 6557 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
21adantr 451 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  =  A )
3 0ss 3517 . . 3  |-  (/)  C_  B
4 0elon 4482 . . . 4  |-  (/)  e.  On
5 oaword 6589 . . . . 5  |-  ( (
(/)  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
653com13 1156 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  (/)  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B
) ) )
74, 6mp3an3 1266 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( (/)  C_  B  <->  ( A  +o  (/) )  C_  ( A  +o  B ) ) )
83, 7mpbii 202 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  (/) )  C_  ( A  +o  B
) )
92, 8eqsstr3d 3247 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    C_ wss 3186   (/)c0 3489   Oncon0 4429  (class class class)co 5900    +o coa 6518
This theorem is referenced by:  oawordexr  6596  oa00  6599  oaf1o  6603  omordi  6606  omeulem2  6623  oeeui  6642  nnarcl  6656  omxpenlem  7006  cantnfle  7417  cantnflem1d  7435  cantnflem3  7438  cantnflem4  7439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-oadd 6525
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