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Theorem ltexpi 8526
Description: Ordering on positive integers in terms of existence of sum. (Contributed by NM, 15-Mar-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexpi  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  E. x  e.  N.  ( A  +N  x )  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltexpi
StepHypRef Expression
1 pinn 8502 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 pinn 8502 . . 3  |-  ( B  e.  N.  ->  B  e.  om )
3 nnaordex 6636 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
41, 2, 3syl2an 463 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
5 ltpiord 8511 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
6 addpiord 8508 . . . . . . 7  |-  ( ( A  e.  N.  /\  x  e.  N. )  ->  ( A  +N  x
)  =  ( A  +o  x ) )
76eqeq1d 2291 . . . . . 6  |-  ( ( A  e.  N.  /\  x  e.  N. )  ->  ( ( A  +N  x )  =  B  <-> 
( A  +o  x
)  =  B ) )
87pm5.32da 622 . . . . 5  |-  ( A  e.  N.  ->  (
( x  e.  N.  /\  ( A  +N  x
)  =  B )  <-> 
( x  e.  N.  /\  ( A  +o  x
)  =  B ) ) )
9 elni2 8501 . . . . . . 7  |-  ( x  e.  N.  <->  ( x  e.  om  /\  (/)  e.  x
) )
109anbi1i 676 . . . . . 6  |-  ( ( x  e.  N.  /\  ( A  +o  x
)  =  B )  <-> 
( ( x  e. 
om  /\  (/)  e.  x
)  /\  ( A  +o  x )  =  B ) )
11 anass 630 . . . . . 6  |-  ( ( ( x  e.  om  /\  (/)  e.  x )  /\  ( A  +o  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
1210, 11bitri 240 . . . . 5  |-  ( ( x  e.  N.  /\  ( A  +o  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
138, 12syl6bb 252 . . . 4  |-  ( A  e.  N.  ->  (
( x  e.  N.  /\  ( A  +N  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
1413rexbidv2 2566 . . 3  |-  ( A  e.  N.  ->  ( E. x  e.  N.  ( A  +N  x
)  =  B  <->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
1514adantr 451 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( E. x  e. 
N.  ( A  +N  x )  =  B  <->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
164, 5, 153bitr4d 276 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  E. x  e.  N.  ( A  +N  x )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   (/)c0 3455   class class class wbr 4023   omcom 4656  (class class class)co 5858    +o coa 6476   N.cnpi 8466    +N cpli 8467    <N clti 8469
This theorem is referenced by:  ltexnq  8599  archnq  8604
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-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-int 3863  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  df-ni 8496  df-pli 8497  df-lti 8499
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