Theorem onsupsucismax 43707

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 43656	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
2		ssorduni 7733	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduninsuc 7794	. . . . 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 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3889  ∪ cuni 4850  Ord word 6322  Oncon0 6323  suc csuc 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-suc 6329

This theorem is referenced by: (None)