Theorem onsupuni 43253

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 7774	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
2	1	impcom 407	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 ∈ On)
3		elssuni 4913	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
4	3	rgen 3053	. . 3 ⊢ ∀𝑦 ∈ 𝐴 𝑦 ⊆ ∪ 𝐴
5		simpl 482	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → 𝐴 ⊆ On)
6	5	sselda 3958	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	2	adantr 480	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → ∪ 𝐴 ∈ On)
8		ontri1 6386	. . . . . 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 5556	. . . . . 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 3161	. . 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 5554	. . . . . 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 4887	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
20		epel 5556	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
21	20	rexbii 3083	. . . . 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 3132	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ On (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥))
25		epweon 7769	. . . 4 ⊢ E We On
26		weso 5645	. . . 4 ⊢ ( E We On → E Or On)
27	25, 26	mp1i 13	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → E Or On)
28	27	eqsup 9468	. 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 1466	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∪ cuni 4883   class class class wbr 5119   E cep 5552   Or wor 5560   We wwe 5605  Oncon0 6352  supcsup 9452

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355  df-on 6356  df-iota 6484  df-riota 7362  df-sup 9454

This theorem is referenced by:  onsupuni2  43254  onsupintrab  43255  limexissup  43305  limexissupab  43307