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 ∧ A ≠ C))

Proof of Theorem eldifsn

Step	Hyp	Ref	Expression
1		eldif 3221	. 2 ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ ¬ A ∈ {C}))
2		elsncg 3755	. . . 4 ⊢ (A ∈ B → (A ∈ {C} ↔ A = C))
3	2	necon3bbid 2550	. . 3 ⊢ (A ∈ B → (¬ A ∈ {C} ↔ A ≠ C))
4	3	pm5.32i 618	. 2 ⊢ ((A ∈ B ∧ ¬ A ∈ {C}) ↔ (A ∈ B ∧ A ≠ C))
5	1, 4	bitri 240	1 ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ A ≠ C))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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