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Mirrors > Home > MPE Home > Th. List > elrel | Structured version Visualization version GIF version |
Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
elrel | ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5688 | . . . 4 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | sselda 3978 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V)) |
4 | elvv 5755 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
5 | 3, 4 | sylib 217 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3946 〈cop 4638 × cxp 5679 Rel wrel 5686 |
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 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
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 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 df-xp 5687 df-rel 5688 |
This theorem is referenced by: eliunxp 5843 elinxp 6027 unielrel 6284 dfpo2 6306 frxp 8139 frxp2 8157 rntpos 8253 gsum2d2lem 19966 funen1cnv 34881 fundmpss 35538 sscoid 35685 elfuns 35687 eliunxp2 47649 |
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