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Theorem intubeu 47696
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 4967 . . . . . . 7 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦)
2 sseq2 4007 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
32ralrab 3688 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐵𝐴𝑥}𝐶𝑦 ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
41, 3bitri 274 . . . . . 6 (𝐶 {𝑥𝐵𝐴𝑥} ↔ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
54biimpri 227 . . . . 5 (∀𝑦𝐵 (𝐴𝑦𝐶𝑦) → 𝐶 {𝑥𝐵𝐴𝑥})
65adantl 480 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 {𝑥𝐵𝐴𝑥})
7 sseq2 4007 . . . . . . 7 (𝑧 = 𝐶 → (𝐴𝑧𝐴𝐶))
8 simpll 763 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶𝐵)
9 simplr 765 . . . . . . 7 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐴𝐶)
107, 8, 9elrabd 3684 . . . . . 6 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑧𝐵𝐴𝑧})
11 sseq2 4007 . . . . . . 7 (𝑧 = 𝑥 → (𝐴𝑧𝐴𝑥))
1211cbvrabv 3440 . . . . . 6 {𝑧𝐵𝐴𝑧} = {𝑥𝐵𝐴𝑥}
1310, 12eleqtrdi 2841 . . . . 5 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 ∈ {𝑥𝐵𝐴𝑥})
14 intss1 4966 . . . . 5 (𝐶 ∈ {𝑥𝐵𝐴𝑥} → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
1513, 14syl 17 . . . 4 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → {𝑥𝐵𝐴𝑥} ⊆ 𝐶)
166, 15eqssd 3998 . . 3 (((𝐶𝐵𝐴𝐶) ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥})
1716expl 456 . 2 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) → 𝐶 = {𝑥𝐵𝐴𝑥}))
18 ssintub 4969 . . . 4 𝐴 {𝑥𝐵𝐴𝑥}
19 sseq2 4007 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶𝐴 {𝑥𝐵𝐴𝑥}))
2018, 19mpbiri 257 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐴𝐶)
21 eqimss 4039 . . . 4 (𝐶 = {𝑥𝐵𝐴𝑥} → 𝐶 {𝑥𝐵𝐴𝑥})
2221, 4sylib 217 . . 3 (𝐶 = {𝑥𝐵𝐴𝑥} → ∀𝑦𝐵 (𝐴𝑦𝐶𝑦))
2320, 22jca 510 . 2 (𝐶 = {𝑥𝐵𝐴𝑥} → (𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)))
2417, 23impbid1 224 1 (𝐶𝐵 → ((𝐴𝐶 ∧ ∀𝑦𝐵 (𝐴𝑦𝐶𝑦)) ↔ 𝐶 = {𝑥𝐵𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059  {crab 3430  wss 3947   cint 4949
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-int 4950
This theorem is referenced by:  ipolubdm  47699  ipolub  47700
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