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Theorem disjprsn 4719
Description: The disjoint intersection of an unordered pair and a singleton. (Contributed by AV, 23-Jan-2021.)
Assertion
Ref Expression
disjprsn ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)

Proof of Theorem disjprsn
StepHypRef Expression
1 dfsn2 4644 . . 3 {𝐶} = {𝐶, 𝐶}
21ineq2i 4225 . 2 ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶})
3 disjpr2 4718 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
43anidms 566 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
52, 4eqtrid 2787 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wne 2938  cin 3962  c0 4339  {csn 4631  {cpr 4633
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634
This theorem is referenced by:  disjtpsn  4720  disjtp2  4721  diftpsn3  4807  funtpg  6623  funcnvtp  6631  hash3tpexb  14530  prodtp  32834  gsumtp  33044
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