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Theorem onsucelsucr 4490
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4512. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6467. (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 2741 . . . 4 (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2 sucexb 4479 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 133 . . 3 (suc 𝐴 ∈ suc 𝐵𝐴 ∈ V)
4 onelss 4370 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴𝐵 → suc 𝐴𝐵))
5 eqimss 3201 . . . . . . . 8 (suc 𝐴 = 𝐵 → suc 𝐴𝐵)
65a1i 9 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴𝐵))
74, 6jaod 712 . . . . . 6 (𝐵 ∈ On → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
87adantl 275 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
9 elsucg 4387 . . . . . . 7 (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
102, 9sylbi 120 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
1110adantr 274 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
12 eloni 4358 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
13 ordelsuc 4487 . . . . . 6 ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
1412, 13sylan2 284 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ suc 𝐴𝐵))
158, 11, 143imtr4d 202 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
1615impancom 258 . . 3 ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴𝐵))
173, 16mpancom 420 . 2 (suc 𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴𝐵))
1817com12 30 1 (𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 703   = wceq 1348  wcel 2141  Vcvv 2730  wss 3121  Ord word 4345  Oncon0 4346  suc csuc 4348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351  df-suc 4354
This theorem is referenced by:  nnsucelsuc  6467
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