![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4673 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
4 | 1, 2 | opex 4226 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V |
5 | 4 | relsn 4728 | . 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 2148 Vcvv 2737 {csn 3591 〈cop 3594 × cxp 4621 Rel wrel 4628 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-opab 4062 df-xp 4629 df-rel 4630 |
This theorem is referenced by: cnvsn 5107 fsn 5684 |
Copyright terms: Public domain | W3C validator |