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Theorem nnaordex 6652
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 4678 . . . . . 6  |-  ( B  e.  om  ->  B  e.  On )
21adantl 452 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  B  e.  On )
3 onelss 4450 . . . . 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 6651 . . . 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 2357 . . . . . . . . 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 4691 . . . . . . . . . . . 12  |-  (/)  e.  om
11 nnaord 6633 . . . . . . . . . . . 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 6618 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
1514adantr 451 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  +o  (/) )  =  A )
1615eleq1d 2362 . . . . . . . . . 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 2668 . . . . 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 2680 . . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   (/)c0 3468   Oncon0 4408   omcom 4672  (class class class)co 5874    +o coa 6492
This theorem is referenced by:  ltexpi  8542
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
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