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Theorem difprsn1 3697
Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
difprsn1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})

Proof of Theorem difprsn1
StepHypRef Expression
1 necom 2411 . 2 (𝐵𝐴𝐴𝐵)
2 df-pr 3568 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
32equncomi 3254 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
43difeq1i 3222 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5 difun2 3474 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
64, 5eqtri 2178 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7 disjsn2 3624 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8 disj3 3447 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
97, 8sylib 121 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
106, 9eqtr4id 2209 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 134 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wne 2327  cdif 3099  cun 3100  cin 3101  c0 3395  {csn 3561  {cpr 3562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568
This theorem is referenced by:  difprsn2  3698
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