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Theorem disjsnxp 39890
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 4801 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 11 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 39889 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43mptru 1660 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wtru 1653  {csn 4334  Disj wdisj 4777   × cxp 5275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-disj 4778  df-opab 4872  df-xp 5283  df-rel 5284
This theorem is referenced by:  sge0xp  41283
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