Theorem unilbss 47690

Description: Superclass of the greatest lower bound. A dual statement of ssintub 4960. (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 4933	. 2 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴)
2		sseq1 3999	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
3	2	elrab 3675	. . 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 3060	1 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴

Syntax hints:   ∈ wcel 2098  {crab 3424   ⊆ wss 3940  ∪ cuni 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-in 3947  df-ss 3957  df-uni 4900

This theorem is referenced by:  unilbeu  47798