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Theorem difprsn1 3749
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 2444 . 2 (𝐵𝐴𝐴𝐵)
2 df-pr 3617 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
32equncomi 3296 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
43difeq1i 3264 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
5 difun2 3517 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
64, 5eqtri 2210 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
7 disjsn2 3673 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
8 disj3 3490 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
97, 8sylib 122 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
106, 9eqtr4id 2241 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 135 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wne 2360  cdif 3141  cun 3142  cin 3143  c0 3437  {csn 3610  {cpr 3611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3616  df-pr 3617
This theorem is referenced by:  difprsn2  3750
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