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Theorem onsucelsucr 4156
Description: Membership is inherited by predecessors. The converse implies excluded middle, as shown at onsucelsucexmid 4172. (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 2541 . . . 4 (suc A suc B → suc A V)
2 sucexb 4146 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A suc BA V)
4 onelss 4047 . . . . . . 7 (B On → (suc A B → suc AB))
5 eqimss 2975 . . . . . . . 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 4064 . . . . . . 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 4035 . . . . . 6 (B On → Ord B)
13 ordelsuc 4154 . . . . . 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 1373   wcel 1375  Vcvv 2533  wss 2895  Ord word 4023  Oncon0 4024  suc csuc 4026
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031
This theorem is referenced by: (None)
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