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Theorem unilbeu 46523
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 3956 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐴𝐶𝐴))
2 simpll 764 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐵)
3 simplr 766 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐴)
41, 2, 3elrabd 3636 . . . . . 6 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑧𝐵𝑧𝐴})
5 sseq1 3956 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
65cbvrabv 3414 . . . . . 6 {𝑧𝐵𝑧𝐴} = {𝑥𝐵𝑥𝐴}
74, 6eleqtrdi 2848 . . . . 5 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑥𝐵𝑥𝐴})
8 elssuni 4883 . . . . 5 (𝐶 ∈ {𝑥𝐵𝑥𝐴} → 𝐶 {𝑥𝐵𝑥𝐴})
97, 8syl 17 . . . 4 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 {𝑥𝐵𝑥𝐴})
10 unissb 4885 . . . . . . 7 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐶)
11 sseq1 3956 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1211ralrab 3640 . . . . . . 7 (∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐶 ↔ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
1310, 12bitri 274 . . . . . 6 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐶 ↔ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
1413biimpri 227 . . . . 5 (∀𝑦𝐵 (𝑦𝐴𝑦𝐶) → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
1514adantl 482 . . . 4 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
169, 15eqssd 3948 . . 3 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴})
1716expl 458 . 2 (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴}))
18 unilbss 46415 . . . 4 {𝑥𝐵𝑥𝐴} ⊆ 𝐴
19 sseq1 3956 . . . 4 (𝐶 = {𝑥𝐵𝑥𝐴} → (𝐶𝐴 {𝑥𝐵𝑥𝐴} ⊆ 𝐴))
2018, 19mpbiri 257 . . 3 (𝐶 = {𝑥𝐵𝑥𝐴} → 𝐶𝐴)
21 eqimss2 3988 . . . 4 (𝐶 = {𝑥𝐵𝑥𝐴} → {𝑥𝐵𝑥𝐴} ⊆ 𝐶)
2221, 13sylib 217 . . 3 (𝐶 = {𝑥𝐵𝑥𝐴} → ∀𝑦𝐵 (𝑦𝐴𝑦𝐶))
2320, 22jca 512 . 2 (𝐶 = {𝑥𝐵𝑥𝐴} → (𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)))
2417, 23impbid1 224 1 (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) ↔ 𝐶 = {𝑥𝐵𝑥𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3062  {crab 3404  wss 3897   cuni 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3405  df-v 3443  df-in 3904  df-ss 3914  df-uni 4851
This theorem is referenced by:  ipoglbdm  46528  ipoglb  46529
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