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Theorem bnj168 29159
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7562. (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 29158 . . . . . . . . 9  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
32anim2i 554 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  ( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
4 r19.42v 2864 . . . . . . . 8  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  <-> 
( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
53, 4sylibr 205 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  n  =  suc  m
) )
6 neeq1 2611 . . . . . . . . . . 11  |-  ( n  =  suc  m  -> 
( n  =/=  1o  <->  suc  m  =/=  1o ) )
76biimpac 474 . . . . . . . . . 10  |-  ( ( n  =/=  1o  /\  n  =  suc  m )  ->  suc  m  =/=  1o )
8 df-1o 6726 . . . . . . . . . . . . 13  |-  1o  =  suc  (/)
98eqeq2i 2448 . . . . . . . . . . . 12  |-  ( suc  m  =  1o  <->  suc  m  =  suc  (/) )
10 nnon 4853 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  On )
11 0elon 4636 . . . . . . . . . . . . 13  |-  (/)  e.  On
12 suc11 4687 . . . . . . . . . . . . 13  |-  ( ( m  e.  On  /\  (/) 
e.  On )  -> 
( suc  m  =  suc  (/)  <->  m  =  (/) ) )
1310, 11, 12sylancl 645 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  ( suc  m  =  suc  (/)  <->  m  =  (/) ) )
149, 13syl5rbb 251 . . . . . . . . . . 11  |-  ( m  e.  om  ->  (
m  =  (/)  <->  suc  m  =  1o ) )
1514necon3bid 2638 . . . . . . . . . 10  |-  ( m  e.  om  ->  (
m  =/=  (/)  <->  suc  m  =/= 
1o ) )
167, 15syl5ibr 214 . . . . . . . . 9  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  m  =/=  (/) ) )
1716ancld 538 . . . . . . . 8  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  ( (
n  =/=  1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) ) )
1817reximia 2813 . . . . . . 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 632 . . . . . . 7  |-  ( ( ( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  ( n  =/= 
1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2120rexbii 2732 . . . . . 6  |-  ( E. m  e.  om  (
( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2219, 21sylib 190 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
23 simpr 449 . . . . 5  |-  ( ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( n  =  suc  m  /\  m  =/=  (/) ) )
2422, 23bnj31 29146 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =  suc  m  /\  m  =/=  (/) ) )
25 df-rex 2713 . . . 4  |-  ( E. m  e.  om  (
n  =  suc  m  /\  m  =/=  (/) )  <->  E. m
( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2624, 25sylib 190 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  om  /\  (
n  =  suc  m  /\  m  =/=  (/) ) ) )
27 simpr 449 . . . . . . 7  |-  ( ( n  =  suc  m  /\  m  =/=  (/) )  ->  m  =/=  (/) )
2827anim2i 554 . . . . . 6  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e. 
om  /\  m  =/=  (/) ) )
291eleq2i 2502 . . . . . . 7  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
30 eldifsn 3929 . . . . . . 7  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
3129, 30bitr2i 243 . . . . . 6  |-  ( ( m  e.  om  /\  m  =/=  (/) )  <->  m  e.  D )
3228, 31sylib 190 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  m  e.  D
)
33 simprl 734 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  n  =  suc  m )
3432, 33jca 520 . . . 4  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e.  D  /\  n  =  suc  m ) )
3534eximi 1586 . . 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 2713 . 2  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
3836, 37sylibr 205 1  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   (/)c0 3630   {csn 3816   Oncon0 4583   suc csuc 4585   omcom 4847   1oc1o 6719
This theorem is referenced by:  bnj600  29352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-1o 6726
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