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Theorem bnj168 28758
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 7306. (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 28757 . . . . . . . . 9  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
32anim2i 552 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  ( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
4 r19.42v 2694 . . . . . . . 8  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  <-> 
( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
53, 4sylibr 203 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  n  =  suc  m
) )
6 neeq1 2454 . . . . . . . . . . 11  |-  ( n  =  suc  m  -> 
( n  =/=  1o  <->  suc  m  =/=  1o ) )
76biimpac 472 . . . . . . . . . 10  |-  ( ( n  =/=  1o  /\  n  =  suc  m )  ->  suc  m  =/=  1o )
8 df-1o 6479 . . . . . . . . . . . . 13  |-  1o  =  suc  (/)
98eqeq2i 2293 . . . . . . . . . . . 12  |-  ( suc  m  =  1o  <->  suc  m  =  suc  (/) )
10 nnon 4662 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  On )
11 0elon 4445 . . . . . . . . . . . . 13  |-  (/)  e.  On
12 suc11 4496 . . . . . . . . . . . . 13  |-  ( ( m  e.  On  /\  (/) 
e.  On )  -> 
( suc  m  =  suc  (/)  <->  m  =  (/) ) )
1310, 11, 12sylancl 643 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  ( suc  m  =  suc  (/)  <->  m  =  (/) ) )
149, 13syl5rbb 249 . . . . . . . . . . 11  |-  ( m  e.  om  ->  (
m  =  (/)  <->  suc  m  =  1o ) )
1514necon3bid 2481 . . . . . . . . . 10  |-  ( m  e.  om  ->  (
m  =/=  (/)  <->  suc  m  =/= 
1o ) )
167, 15syl5ibr 212 . . . . . . . . 9  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  m  =/=  (/) ) )
1716ancld 536 . . . . . . . 8  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  ( (
n  =/=  1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) ) )
1817reximia 2648 . . . . . . 7  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
195, 18syl 15 . . . . . 6  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
20 anass 630 . . . . . . 7  |-  ( ( ( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  ( n  =/= 
1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2120rexbii 2568 . . . . . 6  |-  ( E. m  e.  om  (
( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2219, 21sylib 188 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
23 simpr 447 . . . . 5  |-  ( ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( n  =  suc  m  /\  m  =/=  (/) ) )
2422, 23bnj31 28745 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =  suc  m  /\  m  =/=  (/) ) )
25 df-rex 2549 . . . 4  |-  ( E. m  e.  om  (
n  =  suc  m  /\  m  =/=  (/) )  <->  E. m
( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2624, 25sylib 188 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  om  /\  (
n  =  suc  m  /\  m  =/=  (/) ) ) )
27 simpr 447 . . . . . . 7  |-  ( ( n  =  suc  m  /\  m  =/=  (/) )  ->  m  =/=  (/) )
2827anim2i 552 . . . . . 6  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e. 
om  /\  m  =/=  (/) ) )
291eleq2i 2347 . . . . . . 7  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
30 eldifsn 3749 . . . . . . 7  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
3129, 30bitr2i 241 . . . . . 6  |-  ( ( m  e.  om  /\  m  =/=  (/) )  <->  m  e.  D )
3228, 31sylib 188 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  m  e.  D
)
33 simprl 732 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  n  =  suc  m )
3432, 33jca 518 . . . 4  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e.  D  /\  n  =  suc  m ) )
3534eximi 1563 . . 3  |-  ( E. m ( m  e. 
om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
3626, 35syl 15 . 2  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
37 df-rex 2549 . 2  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
3836, 37sylibr 203 1  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    \ cdif 3149   (/)c0 3455   {csn 3640   Oncon0 4392   suc csuc 4394   omcom 4656   1oc1o 6472
This theorem is referenced by:  bnj600  28951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-1o 6479
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