Theorem intubeu 49481

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 4901	. . . . . 6 ⊢ (𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐶 ⊆ 𝑦)
2		sseq2 3948	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦))
3	2	ralrab 3642	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐶 ⊆ 𝑦 ↔ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
4	1, 3	bitri 276	. . . . 5 ⊢ (𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
5	4	bilanri 507	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
6		sseq2 3948	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝐶))
7		simpll 772	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ 𝐵)
8		simplr 774	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐴 ⊆ 𝐶)
9	6, 7, 8	elrabd 3638	. . . . . 6 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ {𝑧 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑧})
10		sseq2 3948	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝐴 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑥))
11	10	cbvrabv 3402	. . . . . 6 ⊢ {𝑧 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑧} = {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
12	9, 11	eleqtrdi 2850	. . . . 5 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
13		intss1 4900	. . . . 5 ⊢ (𝐶 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐶)
14	12, 13	syl 17	. . . 4 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ⊆ 𝐶)
15	5, 14	eqssd 3939	. . 3 ⊢ (((𝐶 ∈ 𝐵 ∧ 𝐴 ⊆ 𝐶) ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
16	15	expl 458	. 2 ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) → 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
17		ssintub 4903	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}
18		sseq2 3948	. . . 4 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → (𝐴 ⊆ 𝐶 ↔ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
19	17, 18	mpbiri 259	. . 3 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝐶)
20		eqimss 3980	. . . 4 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐶 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥})
21	20, 4	sylib 219	. . 3 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦))
22	19, 21	jca 516	. 2 ⊢ (𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → (𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)))
23	16, 22	impbid1 226	1 ⊢ (𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  {crab 3392   ⊆ wss 3890  ∩ cint 4884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-ss 3907  df-int 4885

This theorem is referenced by:  ipolubdm  49484  ipolub  49485