Theorem dfdif3 3339

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 3228	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		a9ev 1749	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑥
3	2	biantrur 303	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
4		19.41v 1958	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 187	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
6		sb56 1940	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵))
7		equcom 1758	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 238	. . . . . . 7 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵))
9		eleq1w 2299	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
10	9	notbid 677	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵))
11	10	pm5.74i 180	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵))
12		con2b 679	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦))
13		df-ne 2421	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
14	13	bicomi 132	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
15	14	imbi2i 226	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
16	11, 12, 15	3bitri 206	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
17	8, 16	bitri 184	. . . . . 6 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
18	17	albii 1523	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
19	5, 6, 18	3bitri 206	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
20		df-ral 2533	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
21	19, 20	bitr4i 187	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
22	21	rabbii 2808	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
23	1, 22	eqtri 2259	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  {crab 2532   ∖ cdif 3217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ne 2421  df-ral 2533  df-rab 2537  df-dif 3222

This theorem is referenced by: (None)