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Theorem disjsnxp 45010
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 5140 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 11 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 45009 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43mptru 1544 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wtru 1538  {csn 4631  Disj wdisj 5115   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-disj 5116  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  sge0xp  46385
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