Theorem onsupsucismax 44185

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 44134	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
2		ssorduni 7784	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduninsuc 7845	. . . . 5 ⊢ (Ord ∪ 𝐴 → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏))
4	2, 3	syl 18	. . . 4 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = ∪ ∪ 𝐴 ↔ ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏))
5	1, 4	sylibd 242	. . 3 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ¬ ∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏))
6	5	con4d 116	. 2 ⊢ (𝐴 ⊆ On → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))
7	6	adantr 486	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3899  ∪ cuni 4867  Ord word 6358  Oncon0 6359  suc csuc 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6362  df-on 6363  df-suc 6365

This theorem is used by: (None)