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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 2798 | . . 3 ⊢ ∅ = ∅ | |
2 | 1 | orci 862 | . 2 ⊢ (∅ = ∅ ∨ 𝐵 = ∅) |
3 | 0mpo0 7216 | . 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 844 = wceq 1538 ∅c0 4243 ∈ cmpo 7137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-dif 3884 df-nul 4244 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: (None) |
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