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Theorem unilbeu 47000
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 3969 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝐴𝐶𝐴))
2 simpll 765 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐵)
3 simplr 767 . . . . . . 7 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶𝐴)
41, 2, 3elrabd 3647 . . . . . 6 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑧𝐵𝑧𝐴})
5 sseq1 3969 . . . . . . 7 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
65cbvrabv 3417 . . . . . 6 {𝑧𝐵𝑧𝐴} = {𝑥𝐵𝑥𝐴}
74, 6eleqtrdi 2848 . . . . 5 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 ∈ {𝑥𝐵𝑥𝐴})
8 elssuni 4898 . . . . 5 (𝐶 ∈ {𝑥𝐵𝑥𝐴} → 𝐶 {𝑥𝐵𝑥𝐴})
97, 8syl 17 . . . 4 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 {𝑥𝐵𝑥𝐴})
10 unissb 4900 . . . . . . 7 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐶 ↔ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐶)
11 sseq1 3969 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1211ralrab 3651 . . . . . . 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 3961 . . 3 (((𝐶𝐵𝐶𝐴) ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴})
1716expl 458 . 2 (𝐶𝐵 → ((𝐶𝐴 ∧ ∀𝑦𝐵 (𝑦𝐴𝑦𝐶)) → 𝐶 = {𝑥𝐵𝑥𝐴}))
18 unilbss 46892 . . . 4 {𝑥𝐵𝑥𝐴} ⊆ 𝐴
19 sseq1 3969 . . . 4 (𝐶 = {𝑥𝐵𝑥𝐴} → (𝐶𝐴 {𝑥𝐵𝑥𝐴} ⊆ 𝐴))
2018, 19mpbiri 257 . . 3 (𝐶 = {𝑥𝐵𝑥𝐴} → 𝐶𝐴)
21 eqimss2 4001 . . . 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 1541  wcel 2106  wral 3064  {crab 3407  wss 3910   cuni 4865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-in 3917  df-ss 3927  df-uni 4866
This theorem is referenced by:  ipoglbdm  47005  ipoglb  47006
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