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| Mirrors > Home > MPE Home > Th. List > relsnop | Structured version Visualization version 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 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | relsnop.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | relsnopg 5813 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Rel {〈𝐴, 𝐵〉}) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ Rel {〈𝐴, 𝐵〉} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: fsn 7155 imasaddfnlem 17573 ex-res 30460 0funcg 48918 0funcALT 48921 functermc2 49141 | 
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