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Theorem nnarcl 6850
Description: Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
nnarcl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  <->  ( A  e.  om  /\  B  e.  om )
) )

Proof of Theorem nnarcl
StepHypRef Expression
1 oaword1 6786 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
2 eloni 4583 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
3 ordom 4845 . . . . . . 7  |-  Ord  om
4 ordtr2 4617 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( ( A  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  A  e.  om ) )
52, 3, 4sylancl 644 . . . . . 6  |-  ( A  e.  On  ->  (
( A  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  A  e.  om )
)
65exp3a 426 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  A  e.  om ) ) )
76adantr 452 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  A  e.  om )
) )
81, 7mpd 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  A  e.  om )
)
9 oaword2 6787 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
109ancoms 440 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  C_  ( A  +o  B ) )
11 eloni 4583 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
12 ordtr2 4617 . . . . . . 7  |-  ( ( Ord  B  /\  Ord  om )  ->  ( ( B  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  B  e.  om ) )
1311, 3, 12sylancl 644 . . . . . 6  |-  ( B  e.  On  ->  (
( B  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  B  e.  om )
)
1413exp3a 426 . . . . 5  |-  ( B  e.  On  ->  ( B  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  B  e.  om ) ) )
1514adantl 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  B  e.  om )
) )
1610, 15mpd 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  B  e.  om )
)
178, 16jcad 520 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  ( A  e.  om  /\  B  e.  om )
) )
18 nnacl 6845 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
1917, 18impbid1 195 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  <->  ( A  e.  om  /\  B  e.  om )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    C_ wss 3312   Ord word 4572   Oncon0 4573   omcom 4836  (class class class)co 6072    +o coa 6712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-oadd 6719
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