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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 4605 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
2 | relsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | snss 3696 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)) |
4 | 1, 3 | bitr4i 186 | 1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 {csn 3570 × cxp 4596 Rel wrel 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 df-sn 3576 df-rel 4605 |
This theorem is referenced by: relsnop 4704 relsn2m 5068 |
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