Theorem unilbeu 49567

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 3959	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
2		simpll 776	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ 𝐵)
3		simplr 778	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ 𝐴)
4	1, 2, 3	elrabd 3651	. . . . . 6 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴})
5		sseq1 3959	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
6	5	cbvrabv 3423	. . . . . 6 ⊢ {𝑧 ∈ 𝐵 ∣ 𝑧 ⊆ 𝐴} = {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}
7	4, 6	eleqtrdi 2871	. . . . 5 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
8		elssuni 4894	. . . . 5 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
9	7, 8	syl 17	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 ⊆ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
10		unissb 4896	. . . . . 6 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶)
11		sseq1 3959	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
12	11	ralrab 3655	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
13	10, 12	bitri 277	. . . . 5 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
14	13	bilanri 510	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
15	9, 14	eqssd 3951	. . 3 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴})
16	15	expl 461	. 2 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) → 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
17		unilbss 49400	. . . 4 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
18		sseq1 3959	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ↔ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴))
19	17, 18	mpbiri 260	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝐶 ⊆ 𝐴)
20		eqimss2 3993	. . . 4 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐶)
21	20, 13	sylib 220	. . 3 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶))
22	19, 21	jca 519	. 2 ⊢ (𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → (𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)))
23	16, 22	impbid1 227	1 ⊢ (𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   ⊆ wss 3902  ∪ cuni 4862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-ss 3919  df-uni 4863

This theorem is referenced by:  ipoglbdm  49572  ipoglb  49573