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Mirrors > Home > ILE Home > Th. List > invdisj | GIF version |
Description: If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦 ∈ 𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥 ∈ 𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
invdisj | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra2xy 2473 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 | |
2 | df-ral 2419 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥)) | |
3 | rsp 2478 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝐶 = 𝑥)) | |
4 | eqcom 2139 | . . . . . . . . 9 ⊢ (𝐶 = 𝑥 ↔ 𝑥 = 𝐶) | |
5 | 3, 4 | syl6ib 160 | . . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶)) |
6 | 5 | imim2i 12 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 = 𝐶))) |
7 | 6 | impd 252 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
8 | 7 | alimi 1431 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
9 | 2, 8 | sylbi 120 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶)) |
10 | mo2icl 2858 | . . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
12 | 1, 11 | alrimi 1502 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
13 | dfdisj2 3903 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
14 | 12, 13 | sylibr 133 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃*wmo 1998 ∀wral 2414 Disj wdisj 3901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rmo 2422 df-v 2683 df-disj 3902 |
This theorem is referenced by: (None) |
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