Theorem dfdif3OLD 4072

Description: Obsolete version of dfdif3 4071 as of 15-Aug-2025. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dfdif3OLD	⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

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

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfdif3OLD

Step	Hyp	Ref	Expression
1		dfdif2 3913	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵}
2		ax6ev 1988	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑥
3	2	biantrur 538	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
4		19.41v 1968	. . . . . 6 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 280	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵))
6		sbalex 2276	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵))
7		equcom 2037	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 351	. . . . . . 7 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵))
9		eleq1w 2844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
10	9	notbid 320	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵))
11	10	pm5.74i 273	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵))
12		con2b 361	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦))
13		df-ne 2957	. . . . . . . . . 10 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
14	13	bicomi 226	. . . . . . . . 9 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
15	14	imbi2i 338	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
16	11, 12, 15	3bitri 299	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
17	8, 16	bitri 277	. . . . . 6 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
18	17	albii 1838	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
19	5, 6, 18	3bitri 299	. . . 4 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
20		df-ral 3076	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ≠ 𝑦))
21	19, 20	bitr4i 280	. . 3 ⊢ (¬ 𝑥 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦)
22	21	rabbii 3418	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐵} = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
23	1, 22	eqtri 2784	1 ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  {crab 3413   ∖ cdif 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-dif 3907

This theorem is referenced by: (None)