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Theorem unilbss 47502
Description: Superclass of the greatest lower bound. A dual statement of ssintub 4971. (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 4944 . 2 ( {𝑥𝐵𝑥𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥𝐵𝑥𝐴}𝑦𝐴)
2 sseq1 4008 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
32elrab 3684 . . 3 (𝑦 ∈ {𝑥𝐵𝑥𝐴} ↔ (𝑦𝐵𝑦𝐴))
43simprbi 498 . 2 (𝑦 ∈ {𝑥𝐵𝑥𝐴} → 𝑦𝐴)
51, 4mprgbir 3069 1 {𝑥𝐵𝑥𝐴} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {crab 3433  wss 3949   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-uni 4910
This theorem is referenced by:  unilbeu  47610
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