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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 | 0nelopab 5462 | . 2 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 5152 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | eqcomi 2748 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵) |
4 | 3 | eleq2i 2831 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)) |
5 | 1, 4 | mtbi 325 | 1 ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∅c0 4253 {copab 5131 ↦ cmpt 5151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ne 2944 df-v 3425 df-dif 3886 df-un 3888 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5132 df-mpt 5152 |
This theorem is referenced by: (None) |
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