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Theorem difsn 3744
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difsn 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)

Proof of Theorem difsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldifsn 3734 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 109 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 eleq1 2252 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43biimpcd 159 . . . . . . 7 (𝑥𝐵 → (𝑥 = 𝐴𝐴𝐵))
54necon3bd 2403 . . . . . 6 (𝑥𝐵 → (¬ 𝐴𝐵𝑥𝐴))
65com12 30 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
76ancld 325 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
82, 7impbid2 143 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
91, 8bitrid 192 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
109eqrdv 2187 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wcel 2160  wne 2360  cdif 3141  {csn 3607
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-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-dif 3146  df-sn 3613
This theorem is referenced by:  difsnb  3750  fisseneq  6960  dfn2  9219
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