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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 5595 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × V) → 𝐴 ∈ 𝐶) | |
2 | 1 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) |
3 | df-res 5566 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) | |
4 | incom 4177 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) | |
5 | 3, 4 | eqtri 2844 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) |
6 | disjsn 4646 | . . . 4 ⊢ (((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅ ↔ ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) | |
7 | 6 | biimpri 230 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅) |
8 | 5, 7 | syl5eq 2868 | . 2 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
9 | 2, 8 | syl 17 | 1 ⊢ (¬ 𝐴 ∈ 𝐶 → ({〈𝐴, 𝐵〉} ↾ 𝐶) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ∅c0 4290 {csn 4566 〈cop 4572 × cxp 5552 ↾ cres 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-xp 5560 df-res 5566 |
This theorem is referenced by: fvunsn 6940 fsnunres 6949 wfrlem14 7967 ex-res 28219 frrlem12 33134 |
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