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Theorem dfdif3 3260
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 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem dfdif3
StepHypRef Expression
1 dfdif2 3152 . 2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
2 a9ev 1708 . . . . . . 7 𝑦 𝑦 = 𝑥
32biantrur 303 . . . . . 6 𝑥𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
4 19.41v 1914 . . . . . 6 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 187 . . . . 5 𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
6 sb56 1897 . . . . 5 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵))
7 equcom 1717 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
87imbi1i 238 . . . . . . 7 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥𝐵))
9 eleq1w 2250 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
109notbid 668 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝐵 ↔ ¬ 𝑦𝐵))
1110pm5.74i 180 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦𝐵))
12 con2b 670 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑦𝐵) ↔ (𝑦𝐵 → ¬ 𝑥 = 𝑦))
13 df-ne 2361 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
1413bicomi 132 . . . . . . . . 9 𝑥 = 𝑦𝑥𝑦)
1514imbi2i 226 . . . . . . . 8 ((𝑦𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦𝐵𝑥𝑦))
1611, 12, 153bitri 206 . . . . . . 7 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
178, 16bitri 184 . . . . . 6 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
1817albii 1481 . . . . 5 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
195, 6, 183bitri 206 . . . 4 𝑥𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
20 df-ral 2473 . . . 4 (∀𝑦𝐵 𝑥𝑦 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
2119, 20bitr4i 187 . . 3 𝑥𝐵 ↔ ∀𝑦𝐵 𝑥𝑦)
2221rabbii 2738 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
231, 22eqtri 2210 1 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1362   = wceq 1364  wex 1503  wcel 2160  wne 2360  wral 2468  {crab 2472  cdif 3141
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 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ne 2361  df-ral 2473  df-rab 2477  df-dif 3146
This theorem is referenced by: (None)
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