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Theorem nnsucelsuc 6515
Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4525, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4547. (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 2253 . . . 4  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 suceq 4420 . . . . 5  |-  ( x  =  (/)  ->  suc  x  =  suc  (/) )
32eleq2d 2259 . . . 4  |-  ( x  =  (/)  ->  ( suc 
A  e.  suc  x  <->  suc 
A  e.  suc  (/) ) )
41, 3imbi12d 234 . . 3  |-  ( x  =  (/)  ->  ( ( A  e.  x  ->  suc  A  e.  suc  x
)  <->  ( A  e.  (/)  ->  suc  A  e.  suc  (/) ) ) )
5 eleq2 2253 . . . 4  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 suceq 4420 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
76eleq2d 2259 . . . 4  |-  ( x  =  y  ->  ( suc  A  e.  suc  x  <->  suc 
A  e.  suc  y
) )
85, 7imbi12d 234 . . 3  |-  ( x  =  y  ->  (
( A  e.  x  ->  suc  A  e.  suc  x )  <->  ( A  e.  y  ->  suc  A  e.  suc  y ) ) )
9 eleq2 2253 . . . 4  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 suceq 4420 . . . . 5  |-  ( x  =  suc  y  ->  suc  x  =  suc  suc  y )
1110eleq2d 2259 . . . 4  |-  ( x  =  suc  y  -> 
( suc  A  e.  suc  x  <->  suc  A  e.  suc  suc  y ) )
129, 11imbi12d 234 . . 3  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  suc  A  e. 
suc  x )  <->  ( A  e.  suc  y  ->  suc  A  e.  suc  suc  y
) ) )
13 eleq2 2253 . . . 4  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 suceq 4420 . . . . 5  |-  ( x  =  B  ->  suc  x  =  suc  B )
1514eleq2d 2259 . . . 4  |-  ( x  =  B  ->  ( suc  A  e.  suc  x  <->  suc 
A  e.  suc  B
) )
1613, 15imbi12d 234 . . 3  |-  ( x  =  B  ->  (
( A  e.  x  ->  suc  A  e.  suc  x )  <->  ( A  e.  B  ->  suc  A  e.  suc  B ) ) )
17 noel 3441 . . . 4  |-  -.  A  e.  (/)
1817pm2.21i 647 . . 3  |-  ( A  e.  (/)  ->  suc  A  e. 
suc  (/) )
19 elsuci 4421 . . . . . . . 8  |-  ( A  e.  suc  y  -> 
( A  e.  y  \/  A  =  y ) )
2019adantl 277 . . . . . . 7  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( A  e.  y  \/  A  =  y ) )
21 simpl 109 . . . . . . . 8  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  -> 
( A  e.  y  ->  suc  A  e.  suc  y ) )
22 suceq 4420 . . . . . . . . 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 787 . . . . . . 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 2755 . . . . . . . 8  |-  y  e. 
_V
2726sucex 4516 . . . . . . 7  |-  suc  y  e.  _V
2827elsuc2 4425 . . . . . 6  |-  ( suc 
A  e.  suc  suc  y 
<->  ( suc  A  e. 
suc  y  \/  suc  A  =  suc  y ) )
2925, 28sylibr 134 . . . . 5  |-  ( ( ( A  e.  y  ->  suc  A  e.  suc  y )  /\  A  e.  suc  y )  ->  suc  A  e.  suc  suc  y )
3029ex 115 . . . 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 4617 . 2  |-  ( B  e.  om  ->  ( A  e.  B  ->  suc 
A  e.  suc  B
) )
33 nnon 4627 . . 3  |-  ( B  e.  om  ->  B  e.  On )
34 onsucelsucr 4525 . . 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 129 1  |-  ( B  e.  om  ->  ( A  e.  B  <->  suc  A  e. 
suc  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   (/)c0 3437   Oncon0 4381   suc csuc 4383   omcom 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608
This theorem is referenced by:  nnsucsssuc  6516  nntri3or  6517  nnsucuniel  6519  nnaordi  6532  ennnfonelemhom  12465
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