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Theorem difsn 3501
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 3495 . . 3 (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ (𝑥𝐵𝑥𝐴))
2 simpl 102 . . . 4 ((𝑥𝐵𝑥𝐴) → 𝑥𝐵)
3 eleq1 2100 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
43biimpcd 148 . . . . . . 7 (𝑥𝐵 → (𝑥 = 𝐴𝐴𝐵))
54necon3bd 2248 . . . . . 6 (𝑥𝐵 → (¬ 𝐴𝐵𝑥𝐴))
65com12 27 . . . . 5 𝐴𝐵 → (𝑥𝐵𝑥𝐴))
76ancld 308 . . . 4 𝐴𝐵 → (𝑥𝐵 → (𝑥𝐵𝑥𝐴)))
82, 7impbid2 131 . . 3 𝐴𝐵 → ((𝑥𝐵𝑥𝐴) ↔ 𝑥𝐵))
91, 8syl5bb 181 . 2 𝐴𝐵 → (𝑥 ∈ (𝐵 ∖ {𝐴}) ↔ 𝑥𝐵))
109eqrdv 2038 1 𝐴𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97   = wceq 1243  wcel 1393  wne 2204  cdif 2914  {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-sn 3381
This theorem is referenced by:  difsnb  3506  dfn2  8194
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