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

Theorem nnaordex 7289
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 6691 . . . . . 6  |-  ( B  e.  om  ->  B  e.  On )
21adantl 466 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  B  e.  On )
3 onelss 4910 . . . . 5  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
42, 3syl 16 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 nnawordex 7288 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  E. x  e.  om  ( A  +o  x )  =  B ) )
64, 5sylibd 214 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( A  +o  x
)  =  B ) )
7 simplr 755 . . . . . . . . 9  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  A  e.  B
)
8 eleq2 2516 . . . . . . . . 9  |-  ( ( A  +o  x )  =  B  ->  ( A  e.  ( A  +o  x )  <->  A  e.  B ) )
97, 8syl5ibrcom 222 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  A  e.  ( A  +o  x
) ) )
10 peano1 6704 . . . . . . . . . . . 12  |-  (/)  e.  om
11 nnaord 7270 . . . . . . . . . . . 12  |-  ( (
(/)  e.  om  /\  x  e.  om  /\  A  e. 
om )  ->  ( (/) 
e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x ) ) )
1210, 11mp3an1 1312 . . . . . . . . . . 11  |-  ( ( x  e.  om  /\  A  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
1312ancoms 453 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
14 nna0 7255 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
1514adantr 465 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  +o  (/) )  =  A )
1615eleq1d 2512 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( A  +o  (/) )  e.  ( A  +o  x )  <->  A  e.  ( A  +o  x
) ) )
1713, 16bitrd 253 . . . . . . . . 9  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
1817adantlr 714 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
199, 18sylibrd 234 . . . . . . 7  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  (/)  e.  x
) )
2019ancrd 554 . . . . . 6  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2120reximdva 2918 . . . . 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 434 . . . 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 465 . . 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 40 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2517biimpa 484 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  A  e.  ( A  +o  x ) )
2625, 8syl5ibcom 220 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  ( ( A  +o  x )  =  B  ->  A  e.  B ) )
2726expimpd 603 . . . 4  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2827rexlimdva 2935 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2928adantr 465 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. x  e. 
om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
3024, 29impbid 191 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  <->  E. x  e.  om  ( (/) 
e.  x  /\  ( A  +o  x )  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461   (/)c0 3770   Oncon0 4868  (class class class)co 6281   omcom 6685    +o coa 7129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-oadd 7136
This theorem is referenced by:  ltexpi  9283
  Copyright terms: Public domain W3C validator