Theorem onsupuni 43218

Description: The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.)

Assertion

Ref	Expression
onsupuni	⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)

Proof of Theorem onsupuni

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

Step	Hyp	Ref	Expression
1		ssonuni 7799	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
2	1	impcom 407	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 ∈ On)
3		elssuni 4942	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
4	3	rgen 3061	. . 3 ⊢ ∀𝑦 ∈ 𝐴 𝑦 ⊆ ∪ 𝐴
5		simpl 482	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → 𝐴 ⊆ On)
6	5	sselda 3995	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	2	adantr 480	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → ∪ 𝐴 ∈ On)
8		ontri1 6420	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝑦))
9	6, 7, 8	syl2anc 584	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝑦))
10		epel 5592	. . . . . 6 ⊢ (∪ 𝐴 E 𝑦 ↔ ∪ 𝐴 ∈ 𝑦)
11	10	notbii 320	. . . . 5 ⊢ (¬ ∪ 𝐴 E 𝑦 ↔ ¬ ∪ 𝐴 ∈ 𝑦)
12	9, 11	bitr4di 289	. . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 E 𝑦))
13	12	ralbidva 3174	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ ∪ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴 E 𝑦))
14	4, 13	mpbii 233	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴 E 𝑦)
15	2	adantr 480	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → ∪ 𝐴 ∈ On)
16		epelg 5590	. . . . . 6 ⊢ (∪ 𝐴 ∈ On → (𝑦 E ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝐴))
17	15, 16	syl 17	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝐴))
18	17	biimpd 229	. . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴))
19		eluni2 4916	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
20		epel 5592	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
21	20	rexbii 3092	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 E 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
22	19, 21	bitr4i 278	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 E 𝑥)
23	18, 22	imbitrdi 251	. . 3 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥))
24	23	ralrimiva 3144	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ On (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥))
25		epweon 7794	. . . 4 ⊢ E We On
26		weso 5680	. . . 4 ⊢ ( E We On → E Or On)
27	25, 26	mp1i 13	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → E Or On)
28	27	eqsup 9494	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((∪ 𝐴 ∈ On ∧ ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴 E 𝑦 ∧ ∀𝑦 ∈ On (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥)) → sup(𝐴, On, E ) = ∪ 𝐴))
29	2, 14, 24, 28	mp3and 1463	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∪ cuni 4912   class class class wbr 5148   E cep 5588   Or wor 5596   We wwe 5640  Oncon0 6386  supcsup 9478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-iota 6516  df-riota 7388  df-sup 9480

This theorem is referenced by:  onsupuni2  43219  onsupintrab  43220  limexissup  43271  limexissupab  43273