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Theorem eldifsn 3839
 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 3221 . 2 (A (B {C}) ↔ (A B ¬ A {C}))
2 elsncg 3755 . . . 4 (A B → (A {C} ↔ A = C))
32necon3bbid 2550 . . 3 (A B → (¬ A {C} ↔ AC))
43pm5.32i 618 . 2 ((A B ¬ A {C}) ↔ (A B AC))
51, 4bitri 240 1 (A (B {C}) ↔ (A B AC))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741 This theorem is referenced by:  eldifsni  3840  rexdifsn  3843  difsn  3845  nnsucelrlem2  4425  evenfinex  4503  oddfinex  4504  evenoddnnnul  4514  vfinspnn  4541  vfinspsslem1  4550  vinf  4555  enadjlem1  6059  2p1e3c  6156
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