Theorem difsnb 4769

Description: (𝐵 ∖ {𝐴}) equals 𝐵 if and only if 𝐴 is not a member of 𝐵. Generalization of difsn 4761. (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 4761	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) = 𝐵)
2		neldifsnd 4756	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∖ {𝐴}))
3		nelne1 3053	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ (𝐵 ∖ {𝐴})) → 𝐵 ≠ (𝐵 ∖ {𝐴}))
4	2, 3	mpdan 700	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐵 ≠ (𝐵 ∖ {𝐴}))
5	4	necomd 3011	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ≠ 𝐵)
6	5	necon2bi 2986	. 2 ⊢ ((𝐵 ∖ {𝐴}) = 𝐵 → ¬ 𝐴 ∈ 𝐵)
7	1, 6	impbii 212	1 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∖ cdif 3896  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-sn 4585

This theorem is used by:  difsnpss  4770  incexclem  15998  mrieqv2d  17806  mreexmrid  17810  mreexexlem2d  17812  mreexexlem4d  17814  acsfiindd  18720