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Theorem onsupsucismax 42518
Description: If the union of a set of ordinals is a successor ordinal, then that union is the maximum element of the set. This is not a bijection because sets where the maximum element is zero or a limit ordinal exist. Lemma 2.11 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
onsupsucismax ((𝐴 ⊆ On ∧ 𝐴𝑉) → (∃𝑏 ∈ On 𝐴 = suc 𝑏 𝐴𝐴))
Distinct variable group:   𝐴,𝑏
Allowed substitution hint:   𝑉(𝑏)

Proof of Theorem onsupsucismax
StepHypRef Expression
1 onsupnmax 42466 . . . 4 (𝐴 ⊆ On → (¬ 𝐴𝐴 𝐴 = 𝐴))
2 ssorduni 7759 . . . . 5 (𝐴 ⊆ On → Ord 𝐴)
3 orduninsuc 7825 . . . . 5 (Ord 𝐴 → ( 𝐴 = 𝐴 ↔ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
42, 3syl 17 . . . 4 (𝐴 ⊆ On → ( 𝐴 = 𝐴 ↔ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
51, 4sylibd 238 . . 3 (𝐴 ⊆ On → (¬ 𝐴𝐴 → ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
65con4d 115 . 2 (𝐴 ⊆ On → (∃𝑏 ∈ On 𝐴 = suc 𝑏 𝐴𝐴))
76adantr 480 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → (∃𝑏 ∈ On 𝐴 = suc 𝑏 𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3062  wss 3940   cuni 4899  Ord word 6353  Oncon0 6354  suc csuc 6356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358  df-suc 6360
This theorem is referenced by: (None)
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