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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 6653 | . 2 ⊢ (𝐴∅𝐵) = (∅‘〈𝐴, 𝐵〉) | |
2 | 0fv 6227 | . 2 ⊢ (∅‘〈𝐴, 𝐵〉) = ∅ | |
3 | 1, 2 | eqtri 2644 | 1 ⊢ (𝐴∅𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∅c0 3915 〈cop 4183 ‘cfv 5888 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: 2mpt20 6882 el2mpt2csbcl 7250 homarcl 16678 oppglsm 18057 iswwlksnon 26740 iswspthsnon 26741 wwlks2onv 26847 mclsrcl 31458 |
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