Theorem dfdif3 4080

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 3922	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		nelb 3247	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥)
3		necom 3017	. . . 4 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
4	3	ralbii 3117	. . 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 3431	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423   ∖ cdif 3910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-dif 3916

This theorem is referenced by: (None)