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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 7404 | . 2 ⊢ (𝐴∅𝐵) = (∅‘〈𝐴, 𝐵〉) | |
2 | 0fv 6925 | . 2 ⊢ (∅‘〈𝐴, 𝐵〉) = ∅ | |
3 | 1, 2 | eqtri 2752 | 1 ⊢ (𝐴∅𝐵) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∅c0 4314 〈cop 4626 ‘cfv 6533 (class class class)co 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-dm 5676 df-iota 6485 df-fv 6541 df-ov 7404 |
This theorem is referenced by: csbov 7444 2mpo0 7648 el2mpocsbcl 8065 homarcl 17977 oppglsm 19547 iswwlksnon 29531 iswspthsnon 29534 mclsrcl 35007 indthinc 47826 indthincALT 47827 prsthinc 47828 |
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