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Theorem bnj168 28803
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7516. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj168.1  |-  D  =  ( om  \  { (/)
} )
Assertion
Ref Expression
bnj168  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
Distinct variable group:    m, n
Allowed substitution hints:    D( m, n)

Proof of Theorem bnj168
StepHypRef Expression
1 bnj168.1 . . . . . . . . . 10  |-  D  =  ( om  \  { (/)
} )
21bnj158 28802 . . . . . . . . 9  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
32anim2i 553 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  ( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
4 r19.42v 2822 . . . . . . . 8  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  <-> 
( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
53, 4sylibr 204 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  n  =  suc  m
) )
6 neeq1 2575 . . . . . . . . . . 11  |-  ( n  =  suc  m  -> 
( n  =/=  1o  <->  suc  m  =/=  1o ) )
76biimpac 473 . . . . . . . . . 10  |-  ( ( n  =/=  1o  /\  n  =  suc  m )  ->  suc  m  =/=  1o )
8 df-1o 6683 . . . . . . . . . . . . 13  |-  1o  =  suc  (/)
98eqeq2i 2414 . . . . . . . . . . . 12  |-  ( suc  m  =  1o  <->  suc  m  =  suc  (/) )
10 nnon 4810 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  On )
11 0elon 4594 . . . . . . . . . . . . 13  |-  (/)  e.  On
12 suc11 4644 . . . . . . . . . . . . 13  |-  ( ( m  e.  On  /\  (/) 
e.  On )  -> 
( suc  m  =  suc  (/)  <->  m  =  (/) ) )
1310, 11, 12sylancl 644 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  ( suc  m  =  suc  (/)  <->  m  =  (/) ) )
149, 13syl5rbb 250 . . . . . . . . . . 11  |-  ( m  e.  om  ->  (
m  =  (/)  <->  suc  m  =  1o ) )
1514necon3bid 2602 . . . . . . . . . 10  |-  ( m  e.  om  ->  (
m  =/=  (/)  <->  suc  m  =/= 
1o ) )
167, 15syl5ibr 213 . . . . . . . . 9  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  m  =/=  (/) ) )
1716ancld 537 . . . . . . . 8  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  ( (
n  =/=  1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) ) )
1817reximia 2771 . . . . . . 7  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
195, 18syl 16 . . . . . 6  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
20 anass 631 . . . . . . 7  |-  ( ( ( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  ( n  =/= 
1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2120rexbii 2691 . . . . . 6  |-  ( E. m  e.  om  (
( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2219, 21sylib 189 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
23 simpr 448 . . . . 5  |-  ( ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( n  =  suc  m  /\  m  =/=  (/) ) )
2422, 23bnj31 28790 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =  suc  m  /\  m  =/=  (/) ) )
25 df-rex 2672 . . . 4  |-  ( E. m  e.  om  (
n  =  suc  m  /\  m  =/=  (/) )  <->  E. m
( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2624, 25sylib 189 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  om  /\  (
n  =  suc  m  /\  m  =/=  (/) ) ) )
27 simpr 448 . . . . . . 7  |-  ( ( n  =  suc  m  /\  m  =/=  (/) )  ->  m  =/=  (/) )
2827anim2i 553 . . . . . 6  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e. 
om  /\  m  =/=  (/) ) )
291eleq2i 2468 . . . . . . 7  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
30 eldifsn 3887 . . . . . . 7  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
3129, 30bitr2i 242 . . . . . 6  |-  ( ( m  e.  om  /\  m  =/=  (/) )  <->  m  e.  D )
3228, 31sylib 189 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  m  e.  D
)
33 simprl 733 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  n  =  suc  m )
3432, 33jca 519 . . . 4  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e.  D  /\  n  =  suc  m ) )
3534eximi 1582 . . 3  |-  ( E. m ( m  e. 
om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
3626, 35syl 16 . 2  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
37 df-rex 2672 . 2  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
3836, 37sylibr 204 1  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    \ cdif 3277   (/)c0 3588   {csn 3774   Oncon0 4541   suc csuc 4543   omcom 4804   1oc1o 6676
This theorem is referenced by:  bnj600  28996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-1o 6683
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