Theorem onsupmaxb 44225

Description: The union of a class of ordinals is an element is an element of that class if and only if there is a maximum element of that class under the epsilon relation, which is to say that the domain of the restricted epsilon relation is not the whole class. (Contributed by RP, 25-Jan-2025.)

Assertion

Ref	Expression
onsupmaxb	⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))

Proof of Theorem onsupmaxb

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

Step	Hyp	Ref	Expression
1		elirrv 9584	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
2		pm5.501 369	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
3	1, 2	mp1i 14	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
4		elequ1 2152	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑥 ↔ 𝑥 ∈ 𝑥))
5	4	notbid 321	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥))
6		elequ1 2152	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
7	6	rexbidv 3187	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
8	5, 7	bibi12d 348	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
9	3, 8	bitr4d 285	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
10	9	biimpd 232	. . . . 5 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
11	10	spimevw 2018	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
12		ssel 3925	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
14	13	imp 412	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
15		ssel2 3926	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
16	15	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
17		ontri1 6396	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
18	14, 16, 17	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
19	18	ralbidva 3184	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
20		ralnex 3089	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
21	19, 20	bitrdi 290	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
22		unissb 4901	. . . . . . . . . 10 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
23		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ⊆ 𝑥)
24		elssuni 4899	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
25	24	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
26	23, 25	eqssd 3948	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 = 𝑥)
27	22, 26	sylan2br 607	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∪ 𝐴 = 𝑥)
28		dfuni2 4869	. . . . . . . . . . 11 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
29	28	eqeq1i 2766	. . . . . . . . . 10 ⊢ (∪ 𝐴 = 𝑥 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥)
30		eqabcb 2901	. . . . . . . . . 10 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥 ↔ ∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥))
31		bicom 225	. . . . . . . . . . 11 ⊢ ((∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
32	31	albii 1852	. . . . . . . . . 10 ⊢ (∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
33	29, 30, 32	3bitri 300	. . . . . . . . 9 ⊢ (∪ 𝐴 = 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
34	27, 33	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
35		notnotb 318	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 ↔ ¬ ¬ 𝑧 ∈ 𝑥)
36	35	bibi1i 341	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ ¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
37		nbbn 386	. . . . . . . . . . 11 ⊢ ((¬ ¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
38	36, 37	bitri 278	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
39	38	albii 1852	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑧 ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
40		alnex 1814	. . . . . . . . 9 ⊢ (∀𝑧 ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
41	39, 40	bitri 278	. . . . . . . 8 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
42	34, 41	sylib 221	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
43	42	ex 418	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
44	21, 43	sylbird 263	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
45	44	con4d 116	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
46	11, 45	impbid2 229	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
47	46	ralbidva 3184	. 2 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
48		dminxp 6172	. . 3 ⊢ (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦)
49		epel 5554	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
50	49	rexbii 3110	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 E 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
51	50	ralbii 3109	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
52	48, 51	bitri 278	. 2 ⊢ (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
53		ralnex 3089	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
54		exnal 1860	. . . . . 6 ⊢ (∃𝑧 ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
55		nbbn 386	. . . . . . . 8 ⊢ ((¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
56	55	bicomi 227	. . . . . . 7 ⊢ (¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
57	56	exbii 1881	. . . . . 6 ⊢ (∃𝑧 ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
58	54, 57	bitr3i 280	. . . . 5 ⊢ (¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
59	58	ralbii 3109	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
60	53, 59	bitr3i 280	. . 3 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
61		uniel 44203	. . 3 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
62	60, 61	xchnxbir 336	. 2 ⊢ (¬ ∪ 𝐴 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
63	47, 52, 62	3bitr4g 317	1 ⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103   E cep 5550   × cxp 5649  dom cdm 5651  Oncon0 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-reg 9579

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6364  df-on 6365

This theorem is used by: (None)