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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 5807 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Rel {⟨𝐴, 𝐵⟩}) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ Rel {⟨𝐴, 𝐵⟩} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3471 {csn 4630 ⟨cop 4636 Rel wrel 5685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5213 df-xp 5686 df-rel 5687 |
This theorem is referenced by: fsn 7148 imasaddfnlem 17515 ex-res 30269 |
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