Theorem unilbeu 49796

Description: Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)

Assertion

Ref	Expression
unilbeu	⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem unilbeu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3963	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
2		simpll 779	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ 𝐵)
3		simplr 781	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ 𝐴)
4	1, 2, 3	elrabd 3654	. . . . . 6 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴})
5		sseq1 3963	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
6	5	cbvrabv 3428	. . . . . 6 ⊢ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴} = {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}
7	4, 6	eleqtrdi 2875	. . . . 5 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
8		elssuni 4906	. . . . 5 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
9	7, 8	syl 18	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10		unissb 4908	. . . . . 6 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶)
11		sseq1 3963	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12	11	ralrab 3659	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
13	10, 12	bitri 278	. . . . 5 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
14	13	bilanri 512	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
15	9, 14	eqssd 3955	. . 3 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
16	15	expl 463	. 2 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
17		unilbss 49629	. . . 4 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
18		sseq1 3963	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ↔ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴))
19	17, 18	mpbiri 261	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ 𝐴)
20		eqimss2 3997	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
21	20, 13	sylib 221	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
22	19, 21	jca 521	. 2 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)))
23	16, 22	impbid1 228	1 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   ⊆ wss 3906  ∪ cuni 4874

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-ss 3923  df-uni 4875

This theorem is used by:  ipoglbdm  49801  ipoglb  49802