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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 4333 | . 2 ⊢ 〈A, B〉 ∈ (V × V) |
4 | opexgOLD 3956 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 ∈ V) | |
5 | 1, 2, 4 | mp2an 402 | . . 3 ⊢ 〈A, B〉 ∈ V |
6 | 5 | relsn 4386 | . 2 ⊢ (Rel {〈A, B〉} ↔ 〈A, B〉 ∈ (V × V)) |
7 | 3, 6 | mpbir 134 | 1 ⊢ Rel {〈A, B〉} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 Vcvv 2551 {csn 3367 〈cop 3370 × cxp 4286 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: cnvsn 4746 fsn 5278 |
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