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Mirrors > Home > ILE Home > Th. List > ressnop0 | GIF version |
Description: If 𝐴 is not in 𝐶, then the restriction of a singleton of 〈𝐴, 𝐵〉 to 𝐶 is null. (Contributed by Scott Fenton, 15-Apr-2011.) |
Ref | Expression |
---|---|
ressnop0 | ⊢ (¬ 𝐴 ∈ 𝐶 → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp1 4568 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × V) → 𝐴 ∈ 𝐶) | |
2 | 1 | con3i 621 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) |
3 | df-res 4546 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) | |
4 | incom 3263 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) | |
5 | 3, 4 | eqtri 2158 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) |
6 | disjsn 3580 | . . . 4 ⊢ (((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅ ↔ ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) | |
7 | 6 | biimpri 132 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅) |
8 | 5, 7 | syl5eq 2182 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
9 | 2, 8 | syl 14 | 1 ⊢ (¬ 𝐴 ∈ 𝐶 → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∩ cin 3065 ∅c0 3358 {csn 3522 〈cop 3525 × cxp 4532 ↾ cres 4536 |
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-in1 603 ax-in2 604 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 df-res 4546 |
This theorem is referenced by: fvunsng 5607 fsnunres 5615 |
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