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Theorem disjsnxp 6231
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 3996 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 9 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 6230 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43mptru 1362 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff set class
Syntax hints:  wtru 1354  {csn 3591  Disj wdisj 3977   × cxp 4620
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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rmo 2463  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217  df-fv 5219  df-1st 6134
This theorem is referenced by:  fsum2dlemstep  11413  fisumcom2  11417  fprod2dlemstep  11601  fprodcom2fi  11605
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