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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 2735 | . . 3 ⊢ ∅ = ∅ | |
2 | 1 | orci 865 | . 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 847 = wceq 1537 ∅c0 4339 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-dif 3966 df-nul 4340 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: (None) |
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