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Theorem nnaordex 6636
Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordex  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem nnaordex
StepHypRef Expression
1 nnon 4662 . . . . . 6  |-  ( B  e.  om  ->  B  e.  On )
21adantl 452 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  B  e.  On )
3 onelss 4434 . . . . 5  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
42, 3syl 15 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 nnawordex 6635 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  E. x  e.  om  ( A  +o  x )  =  B ) )
64, 5sylibd 205 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( A  +o  x
)  =  B ) )
7 simplr 731 . . . . . . . . 9  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  A  e.  B
)
8 eleq2 2344 . . . . . . . . 9  |-  ( ( A  +o  x )  =  B  ->  ( A  e.  ( A  +o  x )  <->  A  e.  B ) )
97, 8syl5ibrcom 213 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  A  e.  ( A  +o  x
) ) )
10 peano1 4675 . . . . . . . . . . . 12  |-  (/)  e.  om
11 nnaord 6617 . . . . . . . . . . . 12  |-  ( (
(/)  e.  om  /\  x  e.  om  /\  A  e. 
om )  ->  ( (/) 
e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x ) ) )
1210, 11mp3an1 1264 . . . . . . . . . . 11  |-  ( ( x  e.  om  /\  A  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
1312ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
14 nna0 6602 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
1514adantr 451 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  +o  (/) )  =  A )
1615eleq1d 2349 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( A  +o  (/) )  e.  ( A  +o  x )  <->  A  e.  ( A  +o  x
) ) )
1713, 16bitrd 244 . . . . . . . . 9  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
1817adantlr 695 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
199, 18sylibrd 225 . . . . . . 7  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  (/)  e.  x
) )
2019ancrd 537 . . . . . 6  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2120reximdva 2655 . . . . 5  |-  ( ( A  e.  om  /\  A  e.  B )  ->  ( E. x  e. 
om  ( A  +o  x )  =  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2221ex 423 . . . 4  |-  ( A  e.  om  ->  ( A  e.  B  ->  ( E. x  e.  om  ( A  +o  x
)  =  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
2322adantr 451 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  ( E. x  e. 
om  ( A  +o  x )  =  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) ) )
246, 23mpdd 36 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2517biimpa 470 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  A  e.  ( A  +o  x ) )
2625, 8syl5ibcom 211 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  ( ( A  +o  x )  =  B  ->  A  e.  B ) )
2726expimpd 586 . . . 4  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2827rexlimdva 2667 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2928adantr 451 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. x  e. 
om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
3024, 29impbid 183 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   (/)c0 3455   Oncon0 4392   omcom 4656  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  ltexpi  8526
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
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