Theorem onsupmaxb 42732

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 9619	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ 𝑥
2		pm5.501 365	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
3	1, 2	mp1i 13	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
4		elequ1 2105	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑥 ↔ 𝑥 ∈ 𝑥))
5	4	notbid 317	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥))
6		elequ1 2105	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
7	6	rexbidv 3169	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
8	5, 7	bibi12d 344	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑥 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)))
9	3, 8	bitr4d 281	. . . . . 6 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
10	9	biimpd 228	. . . . 5 ⊢ (𝑧 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
11	10	spimevw 1990	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
12		ssel 3965	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
13	12	adantr 479	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 → 𝑦 ∈ On))
14	13	imp 405	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
15		ssel2 3967	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
16	15	adantr 479	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
17		ontri1 6398	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
18	14, 16, 17	syl2anc 582	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
19	18	ralbidva 3166	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
20		ralnex 3062	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
21	19, 20	bitrdi 286	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
22		unissb 4937	. . . . . . . . . 10 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
23		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 ⊆ 𝑥)
24		elssuni 4935	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
25	24	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
26	23, 25	eqssd 3990	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∪ 𝐴 ⊆ 𝑥) → ∪ 𝐴 = 𝑥)
27	22, 26	sylan2br 593	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∪ 𝐴 = 𝑥)
28		dfuni2 4905	. . . . . . . . . . 11 ⊢ ∪ 𝐴 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦}
29	28	eqeq1i 2730	. . . . . . . . . 10 ⊢ (∪ 𝐴 = 𝑥 ↔ {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥)
30		eqabcb 2867	. . . . . . . . . 10 ⊢ ({𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦} = 𝑥 ↔ ∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥))
31		bicom 221	. . . . . . . . . . 11 ⊢ ((∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
32	31	albii 1813	. . . . . . . . . 10 ⊢ (∀𝑧(∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
33	29, 30, 32	3bitri 296	. . . . . . . . 9 ⊢ (∪ 𝐴 = 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
34	27, 33	sylib 217	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
35		notnotb 314	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 ↔ ¬ ¬ 𝑧 ∈ 𝑥)
36	35	bibi1i 337	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ ¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
37		nbbn 382	. . . . . . . . . . 11 ⊢ ((¬ ¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
38	36, 37	bitri 274	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
39	38	albii 1813	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑧 ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
40		alnex 1775	. . . . . . . . 9 ⊢ (∀𝑧 ¬ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
41	39, 40	bitri 274	. . . . . . . 8 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
42	34, 41	sylib 217	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥) → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
43	42	ex 411	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
44	21, 43	sylbird 259	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 → ¬ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
45	44	con4d 115	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦))
46	11, 45	impbid2 225	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
47	46	ralbidva 3166	. 2 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦)))
48		dminxp 6179	. . 3 ⊢ (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦)
49		epel 5579	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
50	49	rexbii 3084	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑥 E 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
51	50	ralbii 3083	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 E 𝑦 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
52	48, 51	bitri 274	. 2 ⊢ (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦)
53		ralnex 3062	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
54		exnal 1821	. . . . . 6 ⊢ (∃𝑧 ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
55		nbbn 382	. . . . . . . 8 ⊢ ((¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
56	55	bicomi 223	. . . . . . 7 ⊢ (¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
57	56	exbii 1842	. . . . . 6 ⊢ (∃𝑧 ¬ (𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
58	54, 57	bitr3i 276	. . . . 5 ⊢ (¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
59	58	ralbii 3083	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
60	53, 59	bitr3i 276	. . 3 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
61		uniel 42710	. . 3 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
62	60, 61	xchnxbir 332	. 2 ⊢ (¬ ∪ 𝐴 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑧(¬ 𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))
63	47, 52, 62	3bitr4g 313	1 ⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3051  ∃wrex 3060   ∩ cin 3938   ⊆ wss 3939  ∪ cuni 4903   class class class wbr 5143   E cep 5575   × cxp 5670  dom cdm 5672  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-reg 9615

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-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  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-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368

This theorem is referenced by: (None)