Theorem eldifsn 3840

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 3222	. 2 ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ ¬ A ∈ {C}))
2		elsncg 3756	. . . 4 ⊢ (A ∈ B → (A ∈ {C} ↔ A = C))
3	2	necon3bbid 2551	. . 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 2517   ∖ cdif 3207  {csn 3738

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-sn 3742

This theorem is referenced by:  eldifsni  3841  rexdifsn  3844  difsn  3846  nnsucelrlem2  4426  evenfinex  4504  oddfinex  4505  evenoddnnnul  4515  vfinspnn  4542  vfinspsslem1  4551  vinf  4556  enadjlem1  6060  2p1e3c  6157