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Mirrors > Home > MPE Home > Th. List > Mathboxes > unilbss | Structured version Visualization version GIF version |
Description: Superclass of the greatest lower bound. A dual statement of ssintub 4892. (Contributed by Zhi Wang, 29-Sep-2024.) |
Ref | Expression |
---|---|
unilbss | ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 4868 | . 2 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) | |
2 | sseq1 3941 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
3 | 2 | elrab 3615 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴)) |
4 | 3 | simprbi 500 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴) |
5 | 1, 4 | mprgbir 3077 | 1 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 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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