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Theorem nnaordex 6569
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 4599 . . . . . 6  |-  ( B  e.  om  ->  B  e.  On )
21adantl 454 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  B  e.  On )
3 onelss 4371 . . . . 5  |-  ( B  e.  On  ->  ( A  e.  B  ->  A 
C_  B ) )
42, 3syl 17 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  A  C_  B )
)
5 nnawordex 6568 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  C_  B  <->  E. x  e.  om  ( A  +o  x )  =  B ) )
64, 5sylibd 207 . . 3  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( A  +o  x
)  =  B ) )
7 simplr 734 . . . . . . . . 9  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  A  e.  B
)
8 eleq2 2317 . . . . . . . . 9  |-  ( ( A  +o  x )  =  B  ->  ( A  e.  ( A  +o  x )  <->  A  e.  B ) )
97, 8syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  A  e.  ( A  +o  x
) ) )
10 peano1 4612 . . . . . . . . . . . 12  |-  (/)  e.  om
11 nnaord 6550 . . . . . . . . . . . 12  |-  ( (
(/)  e.  om  /\  x  e.  om  /\  A  e. 
om )  ->  ( (/) 
e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x ) ) )
1210, 11mp3an1 1269 . . . . . . . . . . 11  |-  ( ( x  e.  om  /\  A  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
1312ancoms 441 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  ( A  +o  (/) )  e.  ( A  +o  x
) ) )
14 nna0 6535 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
1514adantr 453 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  +o  (/) )  =  A )
1615eleq1d 2322 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( A  +o  (/) )  e.  ( A  +o  x )  <->  A  e.  ( A  +o  x
) ) )
1713, 16bitrd 246 . . . . . . . . 9  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
1817adantlr 698 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( (/)  e.  x  <->  A  e.  ( A  +o  x ) ) )
199, 18sylibrd 227 . . . . . . 7  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  (/)  e.  x
) )
2019ancrd 539 . . . . . 6  |-  ( ( ( A  e.  om  /\  A  e.  B )  /\  x  e.  om )  ->  ( ( A  +o  x )  =  B  ->  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2120reximdva 2626 . . . . 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 425 . . . 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 453 . . 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 38 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  e.  B  ->  E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B ) ) )
2517biimpa 472 . . . . . 6  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  A  e.  ( A  +o  x ) )
2625, 8syl5ibcom 213 . . . . 5  |-  ( ( ( A  e.  om  /\  x  e.  om )  /\  (/)  e.  x )  ->  ( ( A  +o  x )  =  B  ->  A  e.  B ) )
2726expimpd 589 . . . 4  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2827rexlimdva 2638 . . 3  |-  ( A  e.  om  ->  ( E. x  e.  om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
2928adantr 453 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( E. x  e. 
om  ( (/)  e.  x  /\  ( A  +o  x
)  =  B )  ->  A  e.  B
) )
3024, 29impbid 185 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517    C_ wss 3094   (/)c0 3397   Oncon0 4329   omcom 4593  (class class class)co 5757    +o coa 6409
This theorem is referenced by:  ltexpi  8459
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-oadd 6416
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