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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-0nelmpt | Structured version Visualization version GIF version |
Description: The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.) |
Ref | Expression |
---|---|
bj-0nelmpt | ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0neqopab 6740 | . 2 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 4763 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | eqcomi 2660 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵) |
4 | 3 | eleq2i 2722 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)) |
5 | 1, 4 | mtbi 311 | 1 ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∅c0 3948 {copab 4745 ↦ cmpt 4762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-opab 4746 df-mpt 4763 |
This theorem is referenced by: (None) |
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