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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 4146 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵)) | |
2 | undif2 4418 | . . 3 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | |
3 | 1, 2 | eleqtrrdi 2923 | . 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 4147 | . . 3 ⊢ ((𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) → 𝜓) ↔ ((𝐶 ∈ 𝐴 → 𝜓) ∧ (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))) | |
7 | 4, 5, 6 | sylanbrc 585 | . 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 2113 ∖ cdif 3926 ∪ cun 3927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 |
This theorem is referenced by: prsrcmpltd 32366 |
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