Theorem difsnb 3851

Description: (B ∖ {A}) equals B if and only if A is not a member of B. Generalization of difsn 3846. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difsnb	⊢ (¬ A ∈ B ↔ (B ∖ {A}) = B)

Proof of Theorem difsnb

Step	Hyp	Ref	Expression
1		difsn 3846	. 2 ⊢ (¬ A ∈ B → (B ∖ {A}) = B)
2		neldifsnd 3843	. . . . 5 ⊢ (A ∈ B → ¬ A ∈ (B ∖ {A}))
3		nelne1 2606	. . . . 5 ⊢ ((A ∈ B ∧ ¬ A ∈ (B ∖ {A})) → B ≠ (B ∖ {A}))
4	2, 3	mpdan 649	. . . 4 ⊢ (A ∈ B → B ≠ (B ∖ {A}))
5	4	necomd 2600	. . 3 ⊢ (A ∈ B → (B ∖ {A}) ≠ B)
6	5	necon2bi 2563	. 2 ⊢ ((B ∖ {A}) = B → ¬ A ∈ B)
7	1, 6	impbii 180	1 ⊢ (¬ A ∈ B ↔ (B ∖ {A}) = B)

Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642   ∈ 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:  difsnpss  3852