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Theorem onsucelsucr 4183
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4199. However, the converse does hold where is a natural number, as seen at nnsucelsuc 5985. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr  On  suc  suc

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2543 . . . 4  suc  suc  suc  _V
2 sucexb 4173 . . . 4  _V  suc  _V
31, 2sylibr 137 . . 3  suc  suc  _V
4 onelss 4073 . . . . . . 7  On  suc  suc  C_
5 eqimss 2974 . . . . . . . 8  suc  suc  C_
65a1i 9 . . . . . . 7  On  suc  suc  C_
74, 6jaod 624 . . . . . 6  On  suc  suc  suc  C_
87adantl 262 . . . . 5  _V  On  suc  suc  suc 
C_
9 elsucg 4090 . . . . . . 7  suc  _V  suc  suc  suc  suc
102, 9sylbi 114 . . . . . 6  _V  suc  suc  suc  suc
1110adantr 261 . . . . 5  _V  On  suc  suc  suc  suc
12 eloni 4061 . . . . . 6  On  Ord
13 ordelsuc 4181 . . . . . 6  _V  Ord  suc  C_
1412, 13sylan2 270 . . . . 5  _V  On  suc  C_
158, 11, 143imtr4d 192 . . . 4  _V  On  suc  suc
1615impancom 247 . . 3  _V  suc  suc  On
173, 16mpancom 401 . 2  suc  suc  On
1817com12 27 1  On  suc  suc
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 616   wceq 1228   wcel 1374   _Vcvv 2535    C_ wss 2894   Ord word 4048   Oncon0 4049   suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057
This theorem is referenced by:  nnsucelsuc  5985
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