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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > expanduniss | Structured version Visualization version GIF version |
Description: Expand ∪ 𝐴 ⊆ 𝐵 to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
Ref | Expression |
---|---|
expanduniss | ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 4937 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
2 | df-ss 3957 | . . 3 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)) | |
3 | 2 | expandral 43791 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
4 | 1, 3 | bitri 274 | 1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1531 ∈ wcel 2098 ∀wral 3051 ⊆ wss 3940 ∪ cuni 4903 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-v 3465 df-ss 3957 df-uni 4904 |
This theorem is referenced by: ismnuprim 43795 |
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