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Theorem disjsnxp 42946
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 5083 . . . 4 Disj 𝑗𝐴 {𝑗}
21a1i 11 . . 3 (⊤ → Disj 𝑗𝐴 {𝑗})
32disjxp1 42945 . 2 (⊤ → Disj 𝑗𝐴 ({𝑗} × 𝐵))
43mptru 1547 1 Disj 𝑗𝐴 ({𝑗} × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wtru 1541  {csn 4573  Disj wdisj 5057   × cxp 5618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rmo 3349  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-disj 5058  df-opab 5155  df-xp 5626  df-rel 5627
This theorem is referenced by:  sge0xp  44312
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