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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 4662 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 class class class wbr 4000 Rel wrel 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-rel 4630 |
This theorem is referenced by: nprrel 4668 vtoclr 4671 opeliunxp2 4763 ideqg 4774 issetid 4777 fvmptss2 5587 opeliunxp2f 6233 brtpos2 6246 brdomg 6742 ctex 6747 isfi 6755 en1uniel 6798 xpdom2 6825 xpdom1g 6827 xpen 6839 isbth 6960 djudom 7086 cc3 7255 aprcl 8590 climcl 11271 climi 11276 climrecl 11313 structex 12454 |
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