![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > relxp | GIF version |
Description: A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
relxp | ⊢ Rel (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 4733 | . 2 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
2 | df-rel 4632 | . 2 ⊢ (Rel (𝐴 × 𝐵) ↔ (𝐴 × 𝐵) ⊆ (V × V)) | |
3 | 1, 2 | mpbir 146 | 1 ⊢ Rel (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2737 ⊆ wss 3129 × cxp 4623 Rel wrel 4630 |
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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-opab 4064 df-xp 4631 df-rel 4632 |
This theorem is referenced by: xpiindim 4763 eliunxp 4765 opeliunxp2 4766 relres 4934 restidsing 4962 codir 5016 qfto 5017 cnvcnv 5080 dfco2 5127 unixpm 5163 ressn 5168 fliftcnv 5793 fliftfun 5794 opeliunxp2f 6236 reltpos 6248 tpostpos 6262 tposfo 6269 tposf 6270 swoer 6560 xpider 6603 erinxp 6606 xpcomf1o 6822 ltrel 8015 lerel 8017 fisumcom2 11439 fprodcom2fi 11627 txuni2 13627 txdis1cn 13649 xmeter 13807 reldvg 14019 |
Copyright terms: Public domain | W3C validator |