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Mirrors > Home > MPE Home > Th. List > 0ov | Structured version Visualization version GIF version |
Description: Operation value of the empty set. (Contributed by AV, 15-May-2021.) |
Ref | Expression |
---|---|
0ov | ⊢ (𝐴∅𝐵) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7138 | . 2 ⊢ (𝐴∅𝐵) = (∅‘〈𝐴, 𝐵〉) | |
2 | 0fv 6684 | . 2 ⊢ (∅‘〈𝐴, 𝐵〉) = ∅ | |
3 | 1, 2 | eqtri 2821 | 1 ⊢ (𝐴∅𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 |
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 ax-nul 5174 ax-pow 5231 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-dm 5529 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: csbov 7178 2mpo0 7374 el2mpocsbcl 7763 homarcl 17280 oppglsm 18759 iswwlksnon 27639 iswspthsnon 27642 mclsrcl 32921 |
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