Theorem onsupnmax 42721

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 3090	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
2		ralnex 3062	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
3	2	rexbii 3084	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
4		ssunib 42713	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
5	4	notbii 319	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
6	1, 3, 5	3bitr4ri 303	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
7		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ On)
8	7	sselda 3972	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
9		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → 𝐴 ⊆ On)
10	9	sselda 3972	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
11	10	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
12		ontri1 6398	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
13	8, 11, 12	syl2anc 582	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
14	13	ralbidva 3166	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
15	14	rexbidva 3167	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
16	6, 15	bitr4id 289	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
17		unielid 42712	. . . . . . . . 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 247	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ∪ 𝐴 ∈ 𝐴))
20	16, 19	sylbid 239	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ 𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ∈ 𝐴))
21	20	con1d 145	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → 𝐴 ⊆ ∪ 𝐴))
22		uniss 4911	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴)
23	21, 22	syl6 35	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴))
24		ssorduni 7779	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
25		orduniss 6461	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
27	26	biantrud 530	. . . . . 6 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴)))
28		eqss 3988	. . . . . 6 ⊢ (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴))
29	27, 28	bitr4di 288	. . . . 5 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
30	29	adantr 479	. . . 4 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
31	23, 30	sylibd 238	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∪ 𝐴 ∈ On) → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
32	31	ex 411	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
33		unon 7832	. . . . 5 ⊢ ∪ On = On
34	33	a1i 11	. . . 4 ⊢ (∪ 𝐴 = On → ∪ On = On)
35		unieq 4914	. . . 4 ⊢ (∪ 𝐴 = On → ∪ ∪ 𝐴 = ∪ On)
36		id 22	. . . 4 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = On)
37	34, 35, 36	3eqtr4rd 2776	. . 3 ⊢ (∪ 𝐴 = On → ∪ 𝐴 = ∪ ∪ 𝐴)
38	37	a1i13 27	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 = On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴)))
39		ordeleqon 7782	. . 3 ⊢ (Ord ∪ 𝐴 ↔ (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
40	24, 39	sylib 217	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ∈ On ∨ ∪ 𝐴 = On))
41	32, 38, 40	mpjaod 858	1 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∀wral 3051  ∃wrex 3060   ⊆ wss 3939  ∪ cuni 4903  Ord word 6363  Oncon0 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370

This theorem is referenced by:  onsupeqnmax  42740  onsupsucismax  42773