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Theorem disjsnxp 6142
Description: The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Assertion
Ref Expression
disjsnxp Disj 𝑗𝐴 ({𝑗} × 𝐵)
Distinct variable group:   𝐴,𝑗
Allowed substitution hint:   𝐵(𝑗)

Proof of Theorem disjsnxp
StepHypRef Expression
1 sndisj 3933 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 9 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 6141 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43mptru 1341 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff set class
Syntax hints:  wtru 1333  {csn 3532  Disj wdisj 3914   × cxp 4545
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rmo 2425  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-disj 3915  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-1st 6046
This theorem is referenced by:  fsum2dlemstep  11235  fisumcom2  11239
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