Theorem unilbss 48799

Description: Superclass of the greatest lower bound. A dual statement of ssintub 4926. (Contributed by Zhi Wang, 29-Sep-2024.)

Assertion

Ref	Expression
unilbss	⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unilbss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unissb 4899	. 2 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴)
2		sseq1 3969	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
3	2	elrab 3656	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴))
4	3	simprbi 496	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴)
5	1, 4	mprgbir 3051	1 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴

Syntax hints:   ∈ wcel 2109  {crab 3402   ⊆ wss 3911  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3403  df-v 3446  df-ss 3928  df-uni 4868

This theorem is referenced by:  unilbeu  48966