Mathbox for Glauco Siliprandi |
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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 41358 | . 2 ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵) |
5 | elssuni 4868 | . 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 398 Ⅎwnf 1784 ∈ wcel 2114 ⊆ wss 3936 ∪ cuni 4838 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-int 4877 |
This theorem is referenced by: (None) |
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