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

Theorem ltexpi 9177
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 9153 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 pinn 9153 . . 3  |-  ( B  e.  N.  ->  B  e.  om )
3 nnaordex 7182 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
41, 2, 3syl2an 477 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
5 ltpiord 9162 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
6 addpiord 9159 . . . . . . 7  |-  ( ( A  e.  N.  /\  x  e.  N. )  ->  ( A  +N  x
)  =  ( A  +o  x ) )
76eqeq1d 2454 . . . . . 6  |-  ( ( A  e.  N.  /\  x  e.  N. )  ->  ( ( A  +N  x )  =  B  <-> 
( A  +o  x
)  =  B ) )
87pm5.32da 641 . . . . 5  |-  ( A  e.  N.  ->  (
( x  e.  N.  /\  ( A  +N  x
)  =  B )  <-> 
( x  e.  N.  /\  ( A  +o  x
)  =  B ) ) )
9 elni2 9152 . . . . . . 7  |-  ( x  e.  N.  <->  ( x  e.  om  /\  (/)  e.  x
) )
109anbi1i 695 . . . . . 6  |-  ( ( x  e.  N.  /\  ( A  +o  x
)  =  B )  <-> 
( ( x  e. 
om  /\  (/)  e.  x
)  /\  ( A  +o  x )  =  B ) )
11 anass 649 . . . . . 6  |-  ( ( ( x  e.  om  /\  (/)  e.  x )  /\  ( A  +o  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
1210, 11bitri 249 . . . . 5  |-  ( ( x  e.  N.  /\  ( A  +o  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
138, 12syl6bb 261 . . . 4  |-  ( A  e.  N.  ->  (
( x  e.  N.  /\  ( A  +N  x
)  =  B )  <-> 
( x  e.  om  /\  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
1413rexbidv2 2859 . . 3  |-  ( A  e.  N.  ->  ( E. x  e.  N.  ( A  +N  x
)  =  B  <->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
1514adantr 465 . 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 285 1  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  E. x  e.  N.  ( A  +N  x )  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2797   (/)c0 3740   class class class wbr 4395  (class class class)co 6195   omcom 6581    +o coa 7022   N.cnpi 9117    +N cpli 9118    <N clti 9120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-oadd 7029  df-ni 9147  df-pli 9148  df-lti 9150
This theorem is referenced by:  ltexnq  9250  archnq  9255
  Copyright terms: Public domain W3C validator