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Theorem unilbeu 48774
Description: Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)
Assertion
Ref Expression
unilbeu (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) ↔ 𝐶 = {𝑥𝐵𝑥𝐴}))
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.
StepHypRef Expression
1 sseq1 4021 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐴𝐶𝐴))
2 simpll 767 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐵)
3 simplr 769 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐴)
41, 2, 3elrabd 3697 . . . . . 6 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑧𝐵𝑧𝐴})
5 sseq1 4021 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
65cbvrabv 3444 . . . . . 6 {𝑧𝐵𝑧𝐴} = {𝑥𝐵𝑥𝐴}
74, 6eleqtrdi 2849 . . . . 5 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑥𝐵𝑥𝐴})
8 elssuni 4942 . . . . 5 (𝐶 ∈ {𝑥𝐵𝑥𝐴} → 𝐶 {𝑥𝐵𝑥𝐴})
97, 8syl 17 . . . 4 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 {𝑥𝐵𝑥𝐴})
10 unissb 4944 . . . . . . 7 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐶)
11 sseq1 4021 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1211ralrab 3702 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐶 ↔ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
1310, 12bitri 275 . . . . . 6 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐶 ↔ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
1413biimpri 228 . . . . 5 (∀𝑦𝐵 (𝑦𝐴𝑦𝐶) → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
1514adantl 481 . . . 4 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
169, 15eqssd 4013 . . 3 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴})
1716expl 457 . 2 (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴}))
18 unilbss 48666 . . . 4 {𝑥𝐵𝑥𝐴} ⊆ 𝐴
19 sseq1 4021 . . . 4 (𝐶 = {𝑥𝐵𝑥𝐴} → (𝐶𝐴 {𝑥𝐵𝑥𝐴} ⊆ 𝐴))
2018, 19mpbiri 258 . . 3 (𝐶 = {𝑥𝐵𝑥𝐴} → 𝐶𝐴)
21 eqimss2 4055 . . . 4 (𝐶 = {𝑥𝐵𝑥𝐴} → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
2221, 13sylib 218 . . 3 (𝐶 = {𝑥𝐵𝑥𝐴} → ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
2320, 22jca 511 . 2 (𝐶 = {𝑥𝐵𝑥𝐴} → (𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)))
2417, 23impbid1 225 1 (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) ↔ 𝐶 = {𝑥𝐵𝑥𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-ss 3980  df-uni 4913
This theorem is referenced by:  ipoglbdm  48779  ipoglb  48780
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