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Theorem onsucelsucr 4280
 Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4301. However, the converse does hold where is a natural number, as seen at nnsucelsuc 6155. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2619 . . . 4
2 sucexb 4269 . . . 4
31, 2sylibr 132 . . 3
4 onelss 4170 . . . . . . 7
5 eqimss 3060 . . . . . . . 8
65a1i 9 . . . . . . 7
74, 6jaod 670 . . . . . 6
87adantl 271 . . . . 5
9 elsucg 4187 . . . . . . 7
102, 9sylbi 119 . . . . . 6
1110adantr 270 . . . . 5
12 eloni 4158 . . . . . 6
13 ordelsuc 4277 . . . . . 6
1412, 13sylan2 280 . . . . 5
158, 11, 143imtr4d 201 . . . 4
1615impancom 256 . . 3
173, 16mpancom 413 . 2
1817com12 30 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   wo 662   wceq 1285   wcel 1434  cvv 2610   wss 2982   word 4145  con0 4146   csuc 4148 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-uni 3622  df-tr 3896  df-iord 4149  df-on 4151  df-suc 4154 This theorem is referenced by:  nnsucelsuc  6155
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