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Theorem onsucelsucr 4199
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4215. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6009. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr (B On → (suc A suc BA B))

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2560 . . . 4 (suc A suc B → suc A V)
2 sucexb 4189 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A suc BA V)
4 onelss 4090 . . . . . . 7 (B On → (suc A B → suc AB))
5 eqimss 2991 . . . . . . . 8 (suc A = B → suc AB)
65a1i 9 . . . . . . 7 (B On → (suc A = B → suc AB))
74, 6jaod 636 . . . . . 6 (B On → ((suc A B suc A = B) → suc AB))
87adantl 262 . . . . 5 ((A V B On) → ((suc A B suc A = B) → suc AB))
9 elsucg 4107 . . . . . . 7 (suc A V → (suc A suc B ↔ (suc A B suc A = B)))
102, 9sylbi 114 . . . . . 6 (A V → (suc A suc B ↔ (suc A B suc A = B)))
1110adantr 261 . . . . 5 ((A V B On) → (suc A suc B ↔ (suc A B suc A = B)))
12 eloni 4078 . . . . . 6 (B On → Ord B)
13 ordelsuc 4197 . . . . . 6 ((A V Ord B) → (A B ↔ suc AB))
1412, 13sylan2 270 . . . . 5 ((A V B On) → (A B ↔ suc AB))
158, 11, 143imtr4d 192 . . . 4 ((A V B On) → (suc A suc BA B))
1615impancom 247 . . 3 ((A V suc A suc B) → (B On → A B))
173, 16mpancom 399 . 2 (suc A suc B → (B On → A B))
1817com12 27 1 (B On → (suc A suc BA B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  wss 2911  Ord word 4065  Oncon0 4066  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by:  nnsucelsuc  6009
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