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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssuniint | Structured version Visualization version GIF version |
Description: Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
ssuniint.x | ⊢ Ⅎ𝑥𝜑 |
ssuniint.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ssuniint.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥) |
Ref | Expression |
---|---|
ssuniint | ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssuniint.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ssuniint.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | ssuniint.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥) | |
4 | 1, 2, 3 | elintd 38271 | . 2 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵) |
5 | elssuni 4403 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 → 𝐴 ⊆ ∪ ∩ 𝐵) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 Ⅎwnf 1699 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-int 4411 |
This theorem is referenced by: (None) |
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