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Theorem nnsucelsuc 6101
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4262, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4283. (Contributed by Jim Kingdon, 25-Aug-2019.)
Assertion
Ref Expression
nnsucelsuc  |-  ( B  e.  om  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )

Proof of Theorem nnsucelsuc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2117 . . . 4  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 suceq 4167 . . . . 5  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
32eleq2d 2123 . . . 4  |-  ( x  =  (/)  ->  ( suc 
A  e.  suc  x  <->  suc 
A  e.  suc  (/) ) )
41, 3imbi12d 227 . . 3  |-  ( x  =  (/)  ->  ( ( A  e.  x  ->  suc  A  e.  suc  x
)  <->  ( A  e.  (/)  ->  suc  A  e.  suc  (/) ) ) )
5 eleq2 2117 . . . 4  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 suceq 4167 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
76eleq2d 2123 . . . 4  |-  ( x  =  y  ->  ( suc  A  e.  suc  x  <->  suc 
A  e.  suc  y
) )
85, 7imbi12d 227 . . 3  |-  ( x  =  y  ->  (
( A  e.  x  ->  suc  A  e.  suc  x )  <->  ( A  e.  y  ->  suc  A  e.  suc  y ) ) )
9 eleq2 2117 . . . 4  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 suceq 4167 . . . . 5  |-  ( x  =  suc  y  ->  suc  x  =  suc  suc  y )
1110eleq2d 2123 . . . 4  |-  ( x  =  suc  y  -> 
( suc  A  e.  suc  x  <->  suc  A  e.  suc  suc  y ) )
129, 11imbi12d 227 . . 3  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  suc  A  e. 
suc  x )  <->  ( A  e.  suc  y  ->  suc  A  e.  suc  suc  y
) ) )
13 eleq2 2117 . . . 4  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 suceq 4167 . . . . 5  |-  ( x  =  B  ->  suc  x  =  suc  B )
1514eleq2d 2123 . . . 4  |-  ( x  =  B  ->  ( suc  A  e.  suc  x  <->  suc 
A  e.  suc  B
) )
1613, 15imbi12d 227 . . 3  |-  ( x  =  B  ->  (
( A  e.  x  ->  suc  A  e.  suc  x )  <->  ( A  e.  B  ->  suc  A  e.  suc  B ) ) )
17 noel 3256 . . . 4  |-  -.  A  e.  (/)
1817pm2.21i 585 . . 3  |-  ( A  e.  (/)  ->  suc  A  e. 
suc  (/) )
19 elsuci 4168 . . . . . . . 8  |-  ( A  e.  suc  y  -> 
( A  e.  y  \/  A  =  y ) )
2019adantl 266 . . . . . . 7  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( A  e.  y  \/  A  =  y ) )
21 simpl 106 . . . . . . . 8  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( A  e.  y  ->  suc  A  e.  suc  y ) )
22 suceq 4167 . . . . . . . . 9  |-  ( A  =  y  ->  suc  A  =  suc  y )
2322a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( A  =  y  ->  suc  A  =  suc  y ) )
2421, 23orim12d 710 . . . . . . 7  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( ( A  e.  y  \/  A  =  y )  ->  ( suc  A  e.  suc  y  \/  suc  A  =  suc  y ) ) )
2520, 24mpd 13 . . . . . 6  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( suc  A  e.  suc  y  \/  suc  A  =  suc  y ) )
26 vex 2577 . . . . . . . 8  |-  y  e. 
_V
2726sucex 4253 . . . . . . 7  |-  suc  y  e.  _V
2827elsuc2 4172 . . . . . 6  |-  ( suc 
A  e.  suc  suc  y 
<->  ( suc  A  e. 
suc  y  \/  suc  A  =  suc  y ) )
2925, 28sylibr 141 . . . . 5  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  ->  suc  A  e.  suc  suc  y )
3029ex 112 . . . 4  |-  ( ( A  e.  y  ->  suc  A  e.  suc  y
)  ->  ( A  e.  suc  y  ->  suc  A  e.  suc  suc  y
) )
3130a1i 9 . . 3  |-  ( y  e.  om  ->  (
( A  e.  y  ->  suc  A  e.  suc  y )  ->  ( A  e.  suc  y  ->  suc  A  e.  suc  suc  y ) ) )
324, 8, 12, 16, 18, 31finds 4351 . 2  |-  ( B  e.  om  ->  ( A  e.  B  ->  suc 
A  e.  suc  B
) )
33 nnon 4360 . . 3  |-  ( B  e.  om  ->  B  e.  On )
34 onsucelsucr 4262 . . 3  |-  ( B  e.  On  ->  ( suc  A  e.  suc  B  ->  A  e.  B ) )
3533, 34syl 14 . 2  |-  ( B  e.  om  ->  ( suc  A  e.  suc  B  ->  A  e.  B ) )
3632, 35impbid 124 1  |-  ( B  e.  om  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   (/)c0 3252   Oncon0 4128   suc csuc 4130   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342
This theorem is referenced by:  nnsucsssuc  6102  nntri3or  6103  nnaordi  6112
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