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Mirrors > Home > MPE Home > Th. List > Mathboxes > prsrcmpltd | Structured version Visualization version GIF version |
Description: If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023.) |
Ref | Expression |
---|---|
prsrcmpltd.1 | ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓)) |
prsrcmpltd.2 | ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓)) |
prsrcmpltd.3 | ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓)) |
prsrcmpltd.4 | ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓)) |
Ref | Expression |
---|---|
prsrcmpltd | ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prsrcmpltd.1 | . . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓)) | |
2 | 1 | expdimp 452 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐴 → 𝜓)) |
3 | prsrcmpltd.2 | . . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓)) | |
4 | 3 | expdimp 452 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓)) |
5 | 2, 4 | srcmpltd 34576 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝜓)) |
6 | 5 | impancom 451 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐴 → 𝜓)) |
7 | prsrcmpltd.3 | . . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓)) | |
8 | 7 | expdimp 452 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐴 → 𝜓)) |
9 | prsrcmpltd.4 | . . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓)) | |
10 | 9 | expdimp 452 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓)) |
11 | 8, 10 | srcmpltd 34576 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐵 → 𝜓)) |
12 | 11 | impancom 451 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓)) |
13 | 6, 12 | srcmpltd 34576 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐵 → 𝜓)) |
14 | 13 | ex 412 | . 2 ⊢ (𝜑 → (𝐷 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝜓))) |
15 | 14 | impcomd 411 | 1 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ∖ cdif 3938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 |
This theorem is referenced by: f1resrcmplf1d 34581 |
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