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Mirrors > Home > ILE Home > Th. List > relsn | GIF version |
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) |
Ref | Expression |
---|---|
relsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
relsn | ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4630 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
2 | relsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | snss 3726 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)) |
4 | 1, 3 | bitr4i 187 | 1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 {csn 3591 × 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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-sn 3597 df-rel 4630 |
This theorem is referenced by: relsnop 4729 relsn2m 5095 |
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