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Theorem disjprsn 4394
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 4334 . . 3 {𝐶} = {𝐶, 𝐶}
21ineq2i 3954 . 2 ({𝐴, 𝐵} ∩ {𝐶}) = ({𝐴, 𝐵} ∩ {𝐶, 𝐶})
3 disjpr2 4392 . . 3 (((𝐴𝐶𝐵𝐶) ∧ (𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
43anidms 680 . 2 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶, 𝐶}) = ∅)
52, 4syl5eq 2806 1 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wne 2932  cin 3714  c0 4058  {csn 4321  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by:  disjtpsn  4395  disjtp2  4396  diftpsn3  4477  funtpg  6103  funcnvtp  6112  prodtp  29903
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