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Theorem onsucelsucr 4234
 Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4255. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6070. (Contributed by Jim Kingdon, 17-Jul-2019.)
Assertion
Ref Expression
onsucelsucr (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))

Proof of Theorem onsucelsucr
StepHypRef Expression
1 elex 2566 . . . 4 (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2 sucexb 4223 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 137 . . 3 (suc 𝐴 ∈ suc 𝐵𝐴 ∈ V)
4 onelss 4124 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴𝐵 → suc 𝐴𝐵))
5 eqimss 2997 . . . . . . . 8 (suc 𝐴 = 𝐵 → suc 𝐴𝐵)
65a1i 9 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴𝐵))
74, 6jaod 637 . . . . . 6 (𝐵 ∈ On → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
87adantl 262 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
9 elsucg 4141 . . . . . . 7 (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
102, 9sylbi 114 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
1110adantr 261 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
12 eloni 4112 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
13 ordelsuc 4231 . . . . . 6 ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
1412, 13sylan2 270 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ suc 𝐴𝐵))
158, 11, 143imtr4d 192 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
1615impancom 247 . . 3 ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴𝐵))
173, 16mpancom 399 . 2 (suc 𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴𝐵))
1817com12 27 1 (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  Ord word 4099  Oncon0 4100  suc csuc 4102 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by:  nnsucelsuc  6070
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