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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 5589 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × V) → 𝐴 ∈ 𝐶) | |
2 | 1 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) |
3 | df-res 5560 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) | |
4 | incom 4175 | . . . 4 ⊢ ({〈𝐴, 𝐵〉} ∩ (𝐶 × V)) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) | |
5 | 3, 4 | eqtri 2841 | . . 3 ⊢ ({〈𝐴, 𝐵〉} ↾ 𝐶) = ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) |
6 | disjsn 4639 | . . . 4 ⊢ (((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅ ↔ ¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V)) | |
7 | 6 | biimpri 229 | . . 3 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ (𝐶 × V) → ((𝐶 × V) ∩ {〈𝐴, 𝐵〉}) = ∅) |
8 | 5, 7 | syl5eq 2865 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ∅c0 4288 {csn 4557 〈cop 4563 × cxp 5546 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-res 5560 |
This theorem is referenced by: fvunsn 6933 fsnunres 6942 wfrlem14 7957 ex-res 28147 frrlem12 33031 |
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