Theorem dfdif3 4066

Description: Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof shortened by SN, 15-Aug-2025.)

Assertion

Ref	Expression
dfdif3	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfdif3

Step	Hyp	Ref	Expression
1		dfdif2 3908	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		nelb 3238	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥)
3		necom 3008	. . . 4 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
4	3	ralbii 3108	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
5	2, 4	bitri 278	. 2 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
6	1, 5	rabbieq 3420	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  {crab 3412   ∖ cdif 3896

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 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-dif 3902

This theorem is used by: (None)