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Theorem bnj168 29326
Description: First-order logic and set theory. Revised to remove dependence on ax-reg 8107. (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 29325 . . . . . . . . 9  |-  ( n  e.  D  ->  E. m  e.  om  n  =  suc  m )
32anim2i 571 . . . . . . . 8  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  ( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
4 r19.42v 2990 . . . . . . . 8  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  <-> 
( n  =/=  1o  /\ 
E. m  e.  om  n  =  suc  m ) )
53, 4sylibr 215 . . . . . . 7  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  n  =  suc  m
) )
6 neeq1 2712 . . . . . . . . . . 11  |-  ( n  =  suc  m  -> 
( n  =/=  1o  <->  suc  m  =/=  1o ) )
76biimpac 488 . . . . . . . . . 10  |-  ( ( n  =/=  1o  /\  n  =  suc  m )  ->  suc  m  =/=  1o )
8 df-1o 7190 . . . . . . . . . . . . 13  |-  1o  =  suc  (/)
98eqeq2i 2447 . . . . . . . . . . . 12  |-  ( suc  m  =  1o  <->  suc  m  =  suc  (/) )
10 nnon 6712 . . . . . . . . . . . . 13  |-  ( m  e.  om  ->  m  e.  On )
11 0elon 5495 . . . . . . . . . . . . 13  |-  (/)  e.  On
12 suc11 5545 . . . . . . . . . . . . 13  |-  ( ( m  e.  On  /\  (/) 
e.  On )  -> 
( suc  m  =  suc  (/)  <->  m  =  (/) ) )
1310, 11, 12sylancl 666 . . . . . . . . . . . 12  |-  ( m  e.  om  ->  ( suc  m  =  suc  (/)  <->  m  =  (/) ) )
149, 13syl5rbb 261 . . . . . . . . . . 11  |-  ( m  e.  om  ->  (
m  =  (/)  <->  suc  m  =  1o ) )
1514necon3bid 2689 . . . . . . . . . 10  |-  ( m  e.  om  ->  (
m  =/=  (/)  <->  suc  m  =/= 
1o ) )
167, 15syl5ibr 224 . . . . . . . . 9  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  m  =/=  (/) ) )
1716ancld 555 . . . . . . . 8  |-  ( m  e.  om  ->  (
( n  =/=  1o  /\  n  =  suc  m
)  ->  ( (
n  =/=  1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) ) )
1817reximia 2898 . . . . . . 7  |-  ( E. m  e.  om  (
n  =/=  1o  /\  n  =  suc  m )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
195, 18syl 17 . . . . . 6  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( ( n  =/= 
1o  /\  n  =  suc  m )  /\  m  =/=  (/) ) )
20 anass 653 . . . . . . 7  |-  ( ( ( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  ( n  =/= 
1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2120rexbii 2934 . . . . . 6  |-  ( E. m  e.  om  (
( n  =/=  1o  /\  n  =  suc  m
)  /\  m  =/=  (/) )  <->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2219, 21sylib 199 . . . . 5  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
23 simpr 462 . . . . 5  |-  ( ( n  =/=  1o  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( n  =  suc  m  /\  m  =/=  (/) ) )
2422, 23bnj31 29313 . . . 4  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  om  ( n  =  suc  m  /\  m  =/=  (/) ) )
25 df-rex 2788 . . . 4  |-  ( E. m  e.  om  (
n  =  suc  m  /\  m  =/=  (/) )  <->  E. m
( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) ) )
2624, 25sylib 199 . . 3  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  om  /\  (
n  =  suc  m  /\  m  =/=  (/) ) ) )
27 simpr 462 . . . . . . 7  |-  ( ( n  =  suc  m  /\  m  =/=  (/) )  ->  m  =/=  (/) )
2827anim2i 571 . . . . . 6  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e. 
om  /\  m  =/=  (/) ) )
291eleq2i 2507 . . . . . . 7  |-  ( m  e.  D  <->  m  e.  ( om  \  { (/) } ) )
30 eldifsn 4128 . . . . . . 7  |-  ( m  e.  ( om  \  { (/)
} )  <->  ( m  e.  om  /\  m  =/=  (/) ) )
3129, 30bitr2i 253 . . . . . 6  |-  ( ( m  e.  om  /\  m  =/=  (/) )  <->  m  e.  D )
3228, 31sylib 199 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  m  e.  D
)
33 simprl 762 . . . . 5  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  n  =  suc  m )
3432, 33jca 534 . . . 4  |-  ( ( m  e.  om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  ( m  e.  D  /\  n  =  suc  m ) )
3534eximi 1703 . . 3  |-  ( E. m ( m  e. 
om  /\  ( n  =  suc  m  /\  m  =/=  (/) ) )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
3626, 35syl 17 . 2  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m ( m  e.  D  /\  n  =  suc  m ) )
37 df-rex 2788 . 2  |-  ( E. m  e.  D  n  =  suc  m  <->  E. m
( m  e.  D  /\  n  =  suc  m ) )
3836, 37sylibr 215 1  |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m  e.  D  n  =  suc  m )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   E.wrex 2783    \ cdif 3439   (/)c0 3767   {csn 4002   Oncon0 5442   suc csuc 5444   omcom 6706   1oc1o 7183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6707  df-1o 7190
This theorem is referenced by:  bnj600  29518
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