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Mirrors > Home > ILE Home > Th. List > relsnop | GIF version |
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
relsn.1 | ⊢ 𝐴 ∈ V |
relsnop.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
relsnop | ⊢ Rel {〈𝐴, 𝐵〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | relsnop.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opelvv 4709 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
4 | 1, 2 | opex 4258 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V |
5 | 4 | relsn 4764 | . 2 ⊢ (Rel {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝐵〉 ∈ (V × V)) |
6 | 3, 5 | mpbir 146 | 1 ⊢ Rel {〈𝐴, 𝐵〉} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2164 Vcvv 2760 {csn 3618 〈cop 3621 × cxp 4657 Rel wrel 4664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-opab 4091 df-xp 4665 df-rel 4666 |
This theorem is referenced by: cnvsn 5148 fsn 5730 |
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