Theorem onsupnmax 44323

Description: If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
onsupnmax	⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Proof of Theorem onsupnmax

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexnal 3114	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2		ralnex 3088	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
3	2	rexbii 3109	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
4		ssunib 44315	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	1, 3, 5	3bitr4ri 307	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
7		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ On)
8	7	sselda 3908	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → 𝐴 ⊆ On)
10	9	sselda 3908	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		ontri1 6397	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
13	8, 11, 12	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
14	13	ralbidva 3183	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
15	14	rexbidva 3184	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
16	6, 15	bitr4id 293	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
17		unielid 44314	. . . . . . . . 9 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
19	18	biimprd 251	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
20	16, 19	sylbid 243	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
21	20	con1d 146	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴))
22		uniss 4874	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴)
23	21, 22	syl6 36	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴))
24		ssorduni 7793	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
25		orduniss 6462	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
26	24, 25	syl 18	. . . . . . 7 ⊢ (𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
27	26	biantrud 541	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴)))
28		eqss 3923	. . . . . 6 ⊢ (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴))
29	27, 28	bitr4di 292	. . . . 5 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
30	29	adantr 486	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
31	23, 30	sylibd 242	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
32	31	ex 418	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
33		unon 7842	. . . . 5 ⊢ ∪ On = On
34	33	a1i 11	. . . 4 ⊢ (∪ 𝐴 = On → ∪ On = On)
35		unieq 4877	. . . 4 ⊢ (∪ 𝐴 = On → ∪ ∪ 𝐴 = ∪ On)
36		id 23	. . . 4 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = On)
37	34, 35, 36	3eqtr4rd 2806	. . 3 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
38	37	a1i13 28	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
39		ordeleqon 7796	. . 3 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
40	24, 39	sylib 221	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
41	32, 38, 40	mpjaod 874	1 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ∪ cuni 4866  Ord word 6361  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-suc 6368

This theorem is used by:  onsupeqnmax  44342  onsupsucismax  44374