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Theorem disjprsn 4623
 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 4553 . . 3 {𝐶} = {𝐶, 𝐶}
21ineq2i 4161 . 2 ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶})
3 disjpr2 4622 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
43anidms 570 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
52, 4syl5eq 2868 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 3007   ∩ cin 3909  ∅c0 4266  {csn 4540  {cpr 4542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ne 3008  df-ral 3131  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-sn 4541  df-pr 4543 This theorem is referenced by:  disjtpsn  4624  disjtp2  4625  diftpsn3  4708  funtpg  6382  funcnvtp  6390  prodtp  30529
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