Theorem onsupnmax 43240

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 3100	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2		ralnex 3072	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
3	2	rexbii 3094	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
4		ssunib 43232	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	1, 3, 5	3bitr4ri 304	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
7		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ On)
8	7	sselda 3983	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → 𝐴 ⊆ On)
10	9	sselda 3983	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		ontri1 6418	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
13	8, 11, 12	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
14	13	ralbidva 3176	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
15	14	rexbidva 3177	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
16	6, 15	bitr4id 290	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
17		unielid 43231	. . . . . . . . 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 248	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
20	16, 19	sylbid 240	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
21	20	con1d 145	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴))
22		uniss 4915	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴)
23	21, 22	syl6 35	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴))
24		ssorduni 7799	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
25		orduniss 6481	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
27	26	biantrud 531	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴)))
28		eqss 3999	. . . . . 6 ⊢ (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴))
29	27, 28	bitr4di 289	. . . . 5 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
30	29	adantr 480	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
31	23, 30	sylibd 239	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
32	31	ex 412	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
33		unon 7851	. . . . 5 ⊢ ∪ On = On
34	33	a1i 11	. . . 4 ⊢ (∪ 𝐴 = On → ∪ On = On)
35		unieq 4918	. . . 4 ⊢ (∪ 𝐴 = On → ∪ ∪ 𝐴 = ∪ On)
36		id 22	. . . 4 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = On)
37	34, 35, 36	3eqtr4rd 2788	. . 3 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
38	37	a1i13 27	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
39		ordeleqon 7802	. . 3 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
40	24, 39	sylib 218	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
41	32, 38, 40	mpjaod 861	1 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∪ cuni 4907  Ord word 6383  Oncon0 6384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by:  onsupeqnmax  43259  onsupsucismax  43292