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Theorem disjsnxp 39059
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 4635 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 11 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 39058 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43trud 1491 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wtru 1482  {csn 4168  Disj wdisj 4611   × cxp 5102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-disj 4612  df-opab 4704  df-xp 5110  df-rel 5111
This theorem is referenced by:  sge0xp  40409
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