ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eldifsn Structured version   GIF version

Theorem eldifsn 3469
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 2904 . 2 (A (B ∖ {𝐶}) ↔ (A B ¬ A {𝐶}))
2 elsncg 3372 . . . 4 (A B → (A {𝐶} ↔ A = 𝐶))
32necon3bbid 2223 . . 3 (A B → (¬ A {𝐶} ↔ A𝐶))
43pm5.32i 430 . 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 1374  wne 2186  cdif 2891  {csn 3350
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-sn 3356
This theorem is referenced by:  eldifsni  3470  rexdifsn  3473  difsn  3475  fnniniseg2  5215  rexsupp  5216  suppssfv  5631  suppssov1  5632  dif1o  5936  elni  6168
  Copyright terms: Public domain W3C validator