Theorem onsupuni 44189

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 7783	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
2	1	impcom 413	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 ∈ On)
3		elssuni 4899	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ⊆ ∪ 𝐴)
4	3	rgen 3079	. . 3 ⊢ ∀𝑦 ∈ 𝐴 𝑦 ⊆ ∪ 𝐴
5		simpl 488	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → 𝐴 ⊆ On)
6	5	sselda 3931	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
7	2	adantr 486	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → ∪ 𝐴 ∈ On)
8		ontri1 6390	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ ∪ 𝐴 ∈ On) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝑦))
9	6, 7, 8	syl2anc 596	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝑦))
10		epel 5554	. . . . . 6 ⊢ (∪ 𝐴 E 𝑦 ↔ ∪ 𝐴 ∈ 𝑦)
11	10	notbii 323	. . . . 5 ⊢ (¬ ∪ 𝐴 E 𝑦 ↔ ¬ ∪ 𝐴 ∈ 𝑦)
12	9, 11	bitr4di 292	. . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ ∪ 𝐴 ↔ ¬ ∪ 𝐴 E 𝑦))
13	12	ralbidva 3184	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ ∪ 𝐴 ↔ ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴 E 𝑦))
14	4, 13	mpbii 236	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐴 ¬ ∪ 𝐴 E 𝑦)
15	2	adantr 486	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → ∪ 𝐴 ∈ On)
16		epelg 5552	. . . . . 6 ⊢ (∪ 𝐴 ∈ On → (𝑦 E ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝐴))
17	15, 16	syl 18	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝐴))
18	17	biimpd 232	. . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 → 𝑦 ∈ ∪ 𝐴))
19		eluni2 4871	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
20		epel 5554	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
21	20	rexbii 3110	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 E 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
22	19, 21	bitr4i 281	. . . 4 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 E 𝑥)
23	18, 22	imbitrdi 254	. . 3 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝑦 ∈ On) → (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥))
24	23	ralrimiva 3155	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ On (𝑦 E ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 E 𝑥))
25		epweon 7778	. . . 4 ⊢ E We On
26		weso 5642	. . . 4 ⊢ ( E We On → E Or On)
27	25, 26	mp1i 14	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → E Or On)
28	27	eqsup 9432	. 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 1493	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∪ cuni 4867   class class class wbr 5103   E cep 5550   Or wor 5558   We wwe 5603  Oncon0 6355  supcsup 9416

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-un 7740

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-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  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-ord 6358  df-on 6359  df-iota 6487  df-riota 7369  df-sup 9418

This theorem is used by:  onsupuni2  44190  onsupintrab  44191  limexissup  44241  limexissupab  44243