Theorem ddifnel 3146

Description: Double complement under universal class. The hypothesis corresponds to stability of membership in 𝐴, which is weaker than decidability (see dcimpstab 793). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3147) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that 𝐴 is a subset of V ∖ (V ∖ 𝐴), see ddifss 3253. (Contributed by Jim Kingdon, 21-Jul-2018.)

Hypothesis

Ref	Expression
ddifnel.1	⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
ddifnel	⊢ (V ∖ (V ∖ 𝐴)) = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem ddifnel

Step	Hyp	Ref	Expression
1		ddifnel.1	. . . 4 ⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴)
2	1	adantl 272	. . 3 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) → 𝑥 ∈ 𝐴)
3		elndif 3139	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ (V ∖ 𝐴))
4		vex 2636	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	jctil 306	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)))
6	2, 5	impbii 125	. 2 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V ∖ 𝐴)) ↔ 𝑥 ∈ 𝐴)
7	6	difeqri 3135	1 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∖ cdif 3010

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015

This theorem is referenced by: (None)