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| Mirrors > Home > MPE Home > Th. List > mpo0v | Structured version Visualization version GIF version | ||
| Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) (Proof shortened by AV, 27-Jan-2024.) |
| Ref | Expression |
|---|---|
| mpo0v | ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ ∅ = ∅ | |
| 2 | 1 | orci 866 | . 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅) |
| 3 | 0mpo0 7516 | . 2 ⊢ ((∅ = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅) | |
| 4 | 2, 3 | ax-mp 5 | 1 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 = wceq 1540 ∅c0 4333 ∈ cmpo 7433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-dif 3954 df-nul 4334 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: (None) |
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