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Theorem onsucelsucr 4519
Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4541. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6506. (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 2760 . . . 4 (suc 𝐴 ∈ suc 𝐵 → suc 𝐴 ∈ V)
2 sucexb 4508 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 134 . . 3 (suc 𝐴 ∈ suc 𝐵𝐴 ∈ V)
4 onelss 4399 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴𝐵 → suc 𝐴𝐵))
5 eqimss 3221 . . . . . . . 8 (suc 𝐴 = 𝐵 → suc 𝐴𝐵)
65a1i 9 . . . . . . 7 (𝐵 ∈ On → (suc 𝐴 = 𝐵 → suc 𝐴𝐵))
74, 6jaod 718 . . . . . 6 (𝐵 ∈ On → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
87adantl 277 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → ((suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵) → suc 𝐴𝐵))
9 elsucg 4416 . . . . . . 7 (suc 𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
102, 9sylbi 121 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
1110adantr 276 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵 ↔ (suc 𝐴𝐵 ∨ suc 𝐴 = 𝐵)))
12 eloni 4387 . . . . . 6 (𝐵 ∈ On → Ord 𝐵)
13 ordelsuc 4516 . . . . . 6 ((𝐴 ∈ V ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
1412, 13sylan2 286 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ suc 𝐴𝐵))
158, 11, 143imtr4d 203 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ On) → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
1615impancom 260 . . 3 ((𝐴 ∈ V ∧ suc 𝐴 ∈ suc 𝐵) → (𝐵 ∈ On → 𝐴𝐵))
173, 16mpancom 422 . 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 104  wb 105  wo 709   = wceq 1363  wcel 2158  Vcvv 2749  wss 3141  Ord word 4374  Oncon0 4375  suc csuc 4377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by:  nnsucelsuc  6506
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