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Theorem elnelun 4191
Description: The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.)
Hypotheses
Ref Expression
elneldisj.e 𝐸 = {𝑠𝐴𝐵𝐶}
elneldisj.n 𝑁 = {𝑠𝐴𝐵𝐶}
Assertion
Ref Expression
elnelun (𝐸𝑁) = 𝐴
Distinct variable group:   𝐴,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   𝐸(𝑠)   𝑁(𝑠)

Proof of Theorem elnelun
StepHypRef Expression
1 elneldisj.e . . 3 𝐸 = {𝑠𝐴𝐵𝐶}
2 elneldisj.n . . . 4 𝑁 = {𝑠𝐴𝐵𝐶}
3 df-nel 3103 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
43rabbii 3398 . . . 4 {𝑠𝐴𝐵𝐶} = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
52, 4eqtri 2849 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
61, 5uneq12i 3992 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝐶} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝐶})
7 rabxm 4188 . 2 𝐴 = ({𝑠𝐴𝐵𝐶} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝐶})
86, 7eqtr4i 2852 1 (𝐸𝑁) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1658  wcel 2166  wnel 3102  {crab 3121  cun 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-nel 3103  df-ral 3122  df-rab 3126  df-v 3416  df-un 3803
This theorem is referenced by:  usgrfilem  26624  cusgrsizeinds  26750  vtxdginducedm1  26841
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