Theorem dfdif3 4057

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 3898	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		nelb 3213	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥)
3		necom 2985	. . . 4 ⊢ (𝑦 ≠ 𝑥 ↔ 𝑥 ≠ 𝑦)
4	3	ralbii 3083	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ≠ 𝑥 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
5	2, 4	bitri 275	. 2 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
6	1, 5	rabbieq 3397	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389   ∖ cdif 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-dif 3892

This theorem is referenced by: (None)