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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsupsucismax | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
onsupsucismax | ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsupnmax 42466 | . . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)) | |
2 | ssorduni 7759 | . . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴) | |
3 | orduninsuc 7825 | . . . . 5 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏)) |
5 | 1, 4 | sylibd 238 | . . 3 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏)) |
6 | 5 | con4d 115 | . 2 ⊢ (𝐴 ⊆ On → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴)) |
7 | 6 | adantr 480 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴)) |
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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