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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 6596 | . 2 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 4645 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | eqcomi 2619 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵) |
4 | 3 | eleq2i 2680 | . 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 1475 ∈ wcel 1977 ∅c0 3874 {copab 4642 ↦ cmpt 4643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-mpt 4645 |
This theorem is referenced by: (None) |
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