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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 4395 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × V) → 𝐴 ∈ 𝐶) | |
2 | 1 | con3i 594 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) |
3 | df-res 4375 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) | |
4 | incom 3158 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) | |
5 | 3, 4 | eqtri 2101 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) |
6 | disjsn 3454 | . . . 4 ⊢ (((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅ ↔ ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) | |
7 | 6 | biimpri 131 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅) |
8 | 5, 7 | syl5eq 2125 | . 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 1284 ∈ wcel 1433 Vcvv 2601 ∩ cin 2972 ∅c0 3251 {csn 3398 〈cop 3401 × cxp 4361 ↾ cres 4365 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 df-res 4375 |
This theorem is referenced by: fvunsng 5378 fsnunres 5385 |
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