ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onsucelsucr Structured version   GIF version

Theorem onsucelsucr 4181
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4197. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 5980. (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 2542 . . . 4 (suc A suc B → suc A V)
2 sucexb 4171 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A suc BA V)
4 onelss 4071 . . . . . . 7 (B On → (suc A B → suc AB))
5 eqimss 2973 . . . . . . . 8 (suc A = B → suc AB)
65a1i 9 . . . . . . 7 (B On → (suc A = B → suc AB))
74, 6jaod 624 . . . . . 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 4088 . . . . . . 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 4059 . . . . . 6 (B On → Ord B)
13 ordelsuc 4179 . . . . . 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 401 . 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 616   = wceq 1228   wcel 1375  Vcvv 2534  wss 2893  Ord word 4046  Oncon0 4047  suc csuc 4049
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 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055
This theorem is referenced by:  nnsucelsuc  5980
  Copyright terms: Public domain W3C validator