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Theorem intubeu 48873
Description: Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
Assertion
Ref Expression
intubeu (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) ↔ 𝐶 = {𝑥𝐵𝐴𝑥}))
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.
StepHypRef Expression
1 ssint 4964 . . . . . . 7 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦)
2 sseq2 4010 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32ralrab 3699 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦 ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
41, 3bitri 275 . . . . . 6 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
54biimpri 228 . . . . 5 (∀𝑦𝐵 (𝐴𝑦𝐶𝑦) → 𝐶 {𝑥𝐵𝐴𝑥})
65adantl 481 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 {𝑥𝐵𝐴𝑥})
7 sseq2 4010 . . . . . . 7 (𝑧 = 𝐶 → (𝐴𝑧𝐴𝐶))
8 simpll 767 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶𝐵)
9 simplr 769 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐴𝐶)
107, 8, 9elrabd 3694 . . . . . 6 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑧𝐵𝐴𝑧})
11 sseq2 4010 . . . . . . 7 (𝑧 = 𝑥 → (𝐴𝑧𝐴𝑥))
1211cbvrabv 3447 . . . . . 6 {𝑧𝐵𝐴𝑧} = {𝑥𝐵𝐴𝑥}
1310, 12eleqtrdi 2851 . . . . 5 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑥𝐵𝐴𝑥})
14 intss1 4963 . . . . 5 (𝐶 ∈ {𝑥𝐵𝐴𝑥} → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
1513, 14syl 17 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
166, 15eqssd 4001 . . 3 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥})
1716expl 457 . 2 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥}))
18 ssintub 4966 . . . 4 𝐴 {𝑥𝐵𝐴𝑥}
19 sseq2 4010 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶𝐴 {𝑥𝐵𝐴𝑥}))
2018, 19mpbiri 258 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐴𝐶)
21 eqimss 4042 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐶 {𝑥𝐵𝐴𝑥})
2221, 4sylib 218 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
2320, 22jca 511 . 2 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)))
2417, 23impbid1 225 1 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) ↔ 𝐶 = {𝑥𝐵𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-ss 3968  df-int 4947
This theorem is referenced by:  ipolubdm  48876  ipolub  48877
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