Theorem dfdif3 3186

Description: Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)

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 3079	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		a9ev 1675	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑥
3	2	biantrur 301	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
4		19.41v 1874	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 186	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
6		sb56 1857	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵))
7		equcom 1682	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 237	. . . . . . 7 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵))
9		eleq1w 2200	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
10	9	notbid 656	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵))
11	10	pm5.74i 179	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵))
12		con2b 658	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦))
13		df-ne 2309	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
14	13	bicomi 131	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
15	14	imbi2i 225	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
16	11, 12, 15	3bitri 205	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
17	8, 16	bitri 183	. . . . . 6 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
18	17	albii 1446	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
19	5, 6, 18	3bitri 205	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
20		df-ral 2421	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
21	19, 20	bitr4i 186	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
22	21	rabbii 2672	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
23	1, 22	eqtri 2160	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  {crab 2420   ∖ cdif 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ne 2309  df-ral 2421  df-rab 2425  df-dif 3073

This theorem is referenced by: (None)