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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 6907 | . 2 ⊢ (𝐴∅𝐵) = (∅‘〈𝐴, 𝐵〉) | |
2 | 0fv 6472 | . 2 ⊢ (∅‘〈𝐴, 𝐵〉) = ∅ | |
3 | 1, 2 | eqtri 2848 | 1 ⊢ (𝐴∅𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∅c0 4143 〈cop 4402 ‘cfv 6122 (class class class)co 6904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-nul 5012 ax-pow 5064 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-dm 5351 df-iota 6085 df-fv 6130 df-ov 6907 |
This theorem is referenced by: 2mpt20 7141 el2mpt2csbcl 7512 homarcl 17029 oppglsm 18407 iswwlksnon 27151 iswspthsnon 27154 mclsrcl 32003 |
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