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Mirrors > Home > MPE Home > Th. List > Mathboxes > srcmpltd | Structured version Visualization version GIF version |
Description: If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.) |
Ref | Expression |
---|---|
srcmpltd.1 | ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝜓)) |
srcmpltd.2 | ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓)) |
Ref | Expression |
---|---|
srcmpltd | ⊢ (𝜑 → (𝐶 ∈ 𝐵 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun2 4200 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵)) | |
2 | undif2 4496 | . . 3 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
3 | 1, 2 | eleqtrrdi 2849 | . 2 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴))) |
4 | srcmpltd.1 | . . 3 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝜓)) | |
5 | srcmpltd.2 | . . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓)) | |
6 | elunant 4201 | . . 3 ⊢ ((𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) → 𝜓) ↔ ((𝐶 ∈ 𝐴 → 𝜓) ∧ (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))) | |
7 | 4, 5, 6 | sylanbrc 582 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) → 𝜓)) |
8 | 3, 7 | syl5 34 | 1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ∖ cdif 3967 ∪ cun 3968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 |
This theorem is referenced by: prsrcmpltd 35049 |
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