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Theorem disjprsn 4220
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 4161 . . 3 {𝐶} = {𝐶, 𝐶}
21ineq2i 3789 . 2 ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶})
3 disjpr2 4218 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
43anidms 676 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
52, 4syl5eq 2667 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wne 2790  cin 3554  c0 3891  {csn 4148  {cpr 4150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by:  disjtpsn  4221  disjtp2  4222  diftpsn3  4301  funtpg  5900  funcnvtp  5909
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