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Theorem nnaordex 6604
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 4634 . . . . . 6  |-  ( B  e.  om  ->  B  e.  On )
21adantl 454 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  B  e.  On )
3 onelss 4406 . . . . 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 6603 . . . 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 2319 . . . . . . . . 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 4647 . . . . . . . . . . . 12  |-  (/)  e.  om
11 nnaord 6585 . . . . . . . . . . . 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 6570 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
1514adantr 453 . . . . . . . . . . 11  |-  ( ( A  e.  om  /\  x  e.  om )  ->  ( A  +o  (/) )  =  A )
1615eleq1d 2324 . . . . . . . . . 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 2630 . . . . 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 2642 . . 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 2519    C_ wss 3127   (/)c0 3430   Oncon0 4364   omcom 4628  (class class class)co 5792    +o coa 6444
This theorem is referenced by:  ltexpi  8494
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-oadd 6451
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