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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 4971. (Contributed by Zhi Wang, 29-Sep-2024.) |
Ref | Expression |
---|---|
unilbss | ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 4944 | . 2 ⊢ (∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴 ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}𝑦 ⊆ 𝐴) | |
2 | sseq1 4008 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
3 | 2 | elrab 3684 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ 𝐴)) |
4 | 3 | simprbi 498 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ 𝐴) |
5 | 1, 4 | mprgbir 3069 | 1 ⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 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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