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Theorem unilbss 45867
Description: Superclass of the greatest lower bound. A dual statement of ssintub 4892. (Contributed by Zhi Wang, 29-Sep-2024.)
Assertion
Ref Expression
unilbss {𝑥𝐵𝑥𝐴} ⊆ 𝐴
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unilbss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 unissb 4868 . 2 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴)
2 sseq1 3941 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
32elrab 3615 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
43simprbi 500 . 2 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
51, 4mprgbir 3077 1 {𝑥𝐵𝑥𝐴} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2111  {crab 3066  wss 3881   cuni 4834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-11 2159  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rab 3071  df-v 3423  df-in 3888  df-ss 3898  df-uni 4835
This theorem is referenced by:  unilbeu  45975
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