Theorem intubeu 46158

Description: Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)

Assertion

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

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

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem intubeu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4892	. . . . . . 7 ⊢ (𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐶 ⊆ 𝑦)
2		sseq2 3943	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
3	2	ralrab 3623	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐶 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
4	1, 3	bitri 274	. . . . . 6 ⊢ (𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
5	4	biimpri 227	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦) → 𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
6	5	adantl 481	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
7		sseq2 3943	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝐶))
8		simpll 763	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ 𝐵)
9		simplr 765	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐴 ⊆ 𝐶)
10	7, 8, 9	elrabd 3619	. . . . . 6 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ {𝑧 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑧})
11		sseq2 3943	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐴 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑥))
12	11	cbvrabv 3416	. . . . . 6 ⊢ {𝑧 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑧} = {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
13	10, 12	eleqtrdi 2849	. . . . 5 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
14		intss1 4891	. . . . 5 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐶)
15	13, 14	syl 17	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐶)
16	6, 15	eqssd 3934	. . 3 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
17	16	expl 457	. 2 ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
18		ssintub 4894	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
19		sseq2 3943	. . . 4 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → (𝐴 ⊆ 𝐶 ↔ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
20	18, 19	mpbiri 257	. . 3 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝐶)
21		eqimss 3973	. . . 4 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
22	21, 4	sylib 217	. . 3 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
23	20, 22	jca 511	. 2 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → (𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)))
24	17, 23	impbid1 224	1 ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-int 4877

This theorem is referenced by:  ipolubdm  46161  ipolub  46162