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Theorem dfdif3 3152
 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 3045 . 2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
2 a9ev 1658 . . . . . . 7 𝑦 𝑦 = 𝑥
32biantrur 299 . . . . . 6 𝑥𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
4 19.41v 1856 . . . . . 6 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 186 . . . . 5 𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
6 sb56 1839 . . . . 5 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵))
7 equcom 1665 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
87imbi1i 237 . . . . . . 7 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥𝐵))
9 eleq1w 2175 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
109notbid 639 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝐵 ↔ ¬ 𝑦𝐵))
1110pm5.74i 179 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦𝐵))
12 con2b 641 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑦𝐵) ↔ (𝑦𝐵 → ¬ 𝑥 = 𝑦))
13 df-ne 2283 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
1413bicomi 131 . . . . . . . . 9 𝑥 = 𝑦𝑥𝑦)
1514imbi2i 225 . . . . . . . 8 ((𝑦𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦𝐵𝑥𝑦))
1611, 12, 153bitri 205 . . . . . . 7 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
178, 16bitri 183 . . . . . 6 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
1817albii 1429 . . . . 5 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
195, 6, 183bitri 205 . . . 4 𝑥𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
20 df-ral 2395 . . . 4 (∀𝑦𝐵 𝑥𝑦 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
2119, 20bitr4i 186 . . 3 𝑥𝐵 ↔ ∀𝑦𝐵 𝑥𝑦)
2221rabbii 2643 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
231, 22eqtri 2135 1 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1312   = wceq 1314  ∃wex 1451   ∈ wcel 1463   ≠ wne 2282  ∀wral 2390  {crab 2394   ∖ cdif 3034 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ne 2283  df-ral 2395  df-rab 2399  df-dif 3039 This theorem is referenced by: (None)
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