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| Mirrors > Home > MPE Home > Th. List > ressnop0 | Structured version Visualization version 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 5727 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × V) → 𝐴 ∈ 𝐶) | |
| 2 | df-res 5697 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) | |
| 3 | incom 4209 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) | |
| 4 | 2, 3 | eqtri 2765 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) |
| 5 | disjsn 4711 | . . . 4 ⊢ (((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅ ↔ ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) | |
| 6 | 5 | biimpri 228 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅) |
| 7 | 4, 6 | eqtrid 2789 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
| 8 | 1, 7 | nsyl5 159 | 1 ⊢ (¬ 𝐴 ∈ 𝐶 → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 {csn 4626 〈cop 4632 × cxp 5683 ↾ cres 5687 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: fvunsn 7199 fsnunres 7208 frrlem12 8322 wfrlem14OLD 8362 ex-res 30460 |
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