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Theorem onsucelsucr 4171
 Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4187. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 5973. (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 2534 . . . 4 (suc A suc B → suc A V)
2 sucexb 4161 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A suc BA V)
4 onelss 4062 . . . . . . 7 (B On → (suc A B → suc AB))
5 eqimss 2965 . . . . . . . 8 (suc A = B → suc AB)
65a1i 9 . . . . . . 7 (B On → (suc A = B → suc AB))
74, 6jaod 621 . . . . . 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 4079 . . . . . . 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 4050 . . . . . 6 (B On → Ord B)
13 ordelsuc 4169 . . . . . 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 613   = wceq 1223   ∈ wcel 1366  Vcvv 2526   ⊆ wss 2885  Ord word 4037  Oncon0 4038  suc csuc 4040 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-un 4108 This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-uni 3544  df-tr 3818  df-iord 4041  df-on 4043  df-suc 4046 This theorem is referenced by:  nnsucelsuc  5973
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