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Mirrors > Home > MPE Home > Th. List > elunant | Structured version Visualization version GIF version |
Description: A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.) |
Ref | Expression |
---|---|
elunant | ⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4084 | . . 3 ⊢ (𝐶 ∈ (𝐴 ∪ 𝐵) ↔ (𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵)) | |
2 | 1 | imbi1i 350 | . 2 ⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵) → 𝜑)) |
3 | jaob 959 | . 2 ⊢ (((𝐶 ∈ 𝐴 ∨ 𝐶 ∈ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑))) | |
4 | 2, 3 | bitri 274 | 1 ⊢ ((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∈ wcel 2106 ∪ cun 3886 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3433 df-un 3893 |
This theorem is referenced by: unss 4119 ralunb 4126 intun 4913 srcmpltd 33051 |
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