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Theorem ltexpi 8542
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 8518 . . 3  |-  ( A  e.  N.  ->  A  e.  om )
2 pinn 8518 . . 3  |-  ( B  e.  N.  ->  B  e.  om )
3 nnaordex 6652 . . 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 8527 . 2  |-  ( ( A  e.  N.  /\  B  e.  N. )  ->  ( A  <N  B  <->  A  e.  B ) )
6 addpiord 8524 . . . . . . 7  |-  ( ( A  e.  N.  /\  x  e.  N. )  ->  ( A  +N  x
)  =  ( A  +o  x ) )
76eqeq1d 2304 . . . . . 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 8517 . . . . . . 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 2579 . . 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 1632    e. wcel 1696   E.wrex 2557   (/)c0 3468   class class class wbr 4039   omcom 4672  (class class class)co 5874    +o coa 6492   N.cnpi 8482    +N cpli 8483    <N clti 8485
This theorem is referenced by:  ltexnq  8615  archnq  8620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499  df-ni 8512  df-pli 8513  df-lti 8515
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