Theorem difsnb 4739

Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4731. (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 4731	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2		neldifsnd 4726	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3		nelne1 3031	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
4	2, 3	mpdan 693	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴}))
5	4	necomd 2989	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
6	5	necon2bi 2964	. 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵)
7	1, 6	impbii 210	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∖ cdif 3880  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-sn 4556

This theorem is referenced by:  difsnpss  4740  incexclem  15792  mrieqv2d  17596  mreexmrid  17600  mreexexlem2d  17602  mreexexlem4d  17604  acsfiindd  18510