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Theorem bnj168 29034
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7552. (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 29033 . . . . . . . . 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 2854 . . . . . . . 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 2606 . . . . . . . . . . 11  |-  ( n  =  suc  m  -> 
( n  =/=  1o  <->  suc  m  =/=  1o ) )
76biimpac 473 . . . . . . . . . 10  |-  ( ( n  =/=  1o  /\  n  =  suc  m )  ->  suc  m  =/=  1o )
8 df-1o 6716 . . . . . . . . . . . . 13  |-  1o  =  suc  (/)
98eqeq2i 2445 . . . . . . . . . . . 12  |-  ( suc  m  =  1o  <->  suc  m  =  suc  (/) )
10 nnon 4843 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  On )
11 0elon 4626 . . . . . . . . . . . . 13  |-  (/)  e.  On
12 suc11 4677 . . . . . . . . . . . . 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 2633 . . . . . . . . . 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 2803 . . . . . . 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 2722 . . . . . 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 29021 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =  suc  m  /\  m  =/=  (/) ) )
25 df-rex 2703 . . . 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 2499 . . . . . . 7  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
30 eldifsn 3919 . . . . . . 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 1585 . . 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 2703 . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309   (/)c0 3620   {csn 3806   Oncon0 4573   suc csuc 4575   omcom 4837   1oc1o 6709
This theorem is referenced by:  bnj600  29227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-1o 6716
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