Theorem onsupsucismax 43804

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 ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))

Distinct variable group:   𝐴,𝑏

Allowed substitution hint:   𝑉(𝑏)

Proof of Theorem onsupsucismax

Step	Hyp	Ref	Expression
1		onsupnmax 43753	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
2		ssorduni 7751	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduninsuc 7812	. . . . 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 241	. . 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 483	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∃wrex 3080   ⊆ wss 3899  ∪ cuni 4859  Ord word 6334  Oncon0 6335  suc csuc 6337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-tr 5202  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-ord 6338  df-on 6339  df-suc 6341

This theorem is referenced by: (None)