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Theorem eldifsn 3486
Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)
Assertion
Ref Expression
eldifsn (A (B ∖ {𝐶}) ↔ (A B A𝐶))

Proof of Theorem eldifsn
StepHypRef Expression
1 eldif 2921 . 2 (A (B ∖ {𝐶}) ↔ (A B ¬ A {𝐶}))
2 elsncg 3389 . . . 4 (A B → (A {𝐶} ↔ A = 𝐶))
32necon3bbid 2239 . . 3 (A B → (¬ A {𝐶} ↔ A𝐶))
43pm5.32i 427 . 2 ((A B ¬ A {𝐶}) ↔ (A B A𝐶))
51, 4bitri 173 1 (A (B ∖ {𝐶}) ↔ (A B A𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   wcel 1390  wne 2201  cdif 2908  {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-sn 3373
This theorem is referenced by:  eldifsni  3487  rexdifsn  3490  difsn  3492  fnniniseg2  5233  rexsupp  5234  suppssfv  5650  suppssov1  5651  dif1o  5960  elni  6292  divvalap  7415  elnnne0  7951  divfnzn  8312
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