MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdif3 Structured version   Visualization version   GIF version

Theorem dfdif3 3871
Description: Alternate definition of class difference. (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 3732 . 2 (𝐴𝐵) = {𝑥𝐴 ∣ ¬ 𝑥𝐵}
2 ax6ev 2059 . . . . . . 7 𝑦 𝑦 = 𝑥
32biantrur 520 . . . . . 6 𝑥𝐵 ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
4 19.41v 2029 . . . . . 6 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ (∃𝑦 𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
53, 4bitr4i 267 . . . . 5 𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵))
6 sb56 2271 . . . . 5 (∃𝑦(𝑦 = 𝑥 ∧ ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵))
7 equcom 2103 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
87imbi1i 338 . . . . . . 7 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑥𝐵))
9 eleq1w 2833 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐵𝑦𝐵))
109notbid 307 . . . . . . . . 9 (𝑥 = 𝑦 → (¬ 𝑥𝐵 ↔ ¬ 𝑦𝐵))
1110pm5.74i 260 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑥 = 𝑦 → ¬ 𝑦𝐵))
12 con2b 348 . . . . . . . 8 ((𝑥 = 𝑦 → ¬ 𝑦𝐵) ↔ (𝑦𝐵 → ¬ 𝑥 = 𝑦))
13 df-ne 2944 . . . . . . . . . 10 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
1413bicomi 214 . . . . . . . . 9 𝑥 = 𝑦𝑥𝑦)
1514imbi2i 325 . . . . . . . 8 ((𝑦𝐵 → ¬ 𝑥 = 𝑦) ↔ (𝑦𝐵𝑥𝑦))
1611, 12, 153bitri 286 . . . . . . 7 ((𝑥 = 𝑦 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
178, 16bitri 264 . . . . . 6 ((𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ (𝑦𝐵𝑥𝑦))
1817albii 1895 . . . . 5 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝐵) ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
195, 6, 183bitri 286 . . . 4 𝑥𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
20 df-ral 3066 . . . 4 (∀𝑦𝐵 𝑥𝑦 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
2119, 20bitr4i 267 . . 3 𝑥𝐵 ↔ ∀𝑦𝐵 𝑥𝑦)
2221rabbii 3335 . 2 {𝑥𝐴 ∣ ¬ 𝑥𝐵} = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
231, 22eqtri 2793 1 (𝐴𝐵) = {𝑥𝐴 ∣ ∀𝑦𝐵 𝑥𝑦}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  {crab 3065  cdif 3720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ne 2944  df-ral 3066  df-rab 3070  df-dif 3726
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator