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Mirrors > Home > ILE Home > Th. List > brrelex1i | GIF version |
Description: The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.) |
Ref | Expression |
---|---|
brrelexi.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
brrelex1i | ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brrelexi.1 | . 2 ⊢ Rel 𝑅 | |
2 | brrelex1 4578 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
3 | 1, 2 | mpan 420 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Vcvv 2686 class class class wbr 3929 Rel wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: nprrel 4584 vtoclr 4587 opeliunxp2 4679 ideqg 4690 issetid 4693 fvmptss2 5496 opeliunxp2f 6135 brtpos2 6148 brdomg 6642 ctex 6647 isfi 6655 en1uniel 6698 xpdom2 6725 xpdom1g 6727 xpen 6739 isbth 6855 djudom 6978 aprcl 8408 climcl 11051 climi 11056 climrecl 11093 structex 11971 |
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