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Theorem disjprsn 4714
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 4639 . . 3 {𝐶} = {𝐶, 𝐶}
21ineq2i 4217 . 2 ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶})
3 disjpr2 4713 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
43anidms 566 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
52, 4eqtrid 2789 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wne 2940  cin 3950  c0 4333  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629
This theorem is referenced by:  disjtpsn  4715  disjtp2  4716  diftpsn3  4802  funtpg  6621  funcnvtp  6629  hash3tpexb  14533  prodtp  32829  gsumtp  33061
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