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Theorem elnelunOLD 4110
Description: Obsolete version of elnelun 4108 as of 17-Dec-2021. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elneldisjOLD.e 𝐸 = {𝑠𝐴𝐵𝑠}
elneldisjOLD.f 𝑁 = {𝑠𝐴𝐵𝑠}
Assertion
Ref Expression
elnelunOLD (𝐸𝑁) = 𝐴
Distinct variable group:   𝐴,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elnelunOLD
StepHypRef Expression
1 elneldisjOLD.e . . 3 𝐸 = {𝑠𝐴𝐵𝑠}
2 elneldisjOLD.f . . . 4 𝑁 = {𝑠𝐴𝐵𝑠}
3 df-nel 3047 . . . . . 6 (𝐵𝑠 ↔ ¬ 𝐵𝑠)
43a1i 11 . . . . 5 (𝑠𝐴 → (𝐵𝑠 ↔ ¬ 𝐵𝑠))
54rabbiia 3334 . . . 4 {𝑠𝐴𝐵𝑠} = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
62, 5eqtri 2793 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝑠}
71, 6uneq12i 3916 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
8 rabxm 4105 . 2 𝐴 = ({𝑠𝐴𝐵𝑠} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝑠})
97, 8eqtr4i 2796 1 (𝐸𝑁) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1631  wcel 2145  wnel 3046  {crab 3065  cun 3721
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-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-nel 3047  df-ral 3066  df-rab 3070  df-v 3353  df-un 3728
This theorem is referenced by: (None)
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