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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 | . . 3 ⊢ 𝐴 ∈ V | |
2 | relsnop.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opelvv 5088 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
4 | opex 4859 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
5 | 4 | relsn 5146 | . 2 ⊢ (Rel {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝐵〉 ∈ (V × V)) |
6 | 3, 5 | mpbir 220 | 1 ⊢ Rel {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 × cxp 5036 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: cnvsn 5536 fsn 6308 imasaddfnlem 16011 ex-res 26690 |
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