Theorem onsupsucismax 43382

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 43331	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
2		ssorduni 7712	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduninsuc 7773	. . . . 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 239	. . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897  ∪ cuni 4856  Ord word 6305  Oncon0 6306  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310  df-suc 6312

This theorem is referenced by: (None)