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Theorem elnelun 4347
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 3129 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
43rabbii 3479 . . . 4 {𝑠𝐴𝐵𝐶} = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
52, 4eqtri 2849 . . 3 𝑁 = {𝑠𝐴 ∣ ¬ 𝐵𝐶}
61, 5uneq12i 4141 . 2 (𝐸𝑁) = ({𝑠𝐴𝐵𝐶} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝐶})
7 rabxm 4344 . 2 𝐴 = ({𝑠𝐴𝐵𝐶} ∪ {𝑠𝐴 ∣ ¬ 𝐵𝐶})
86, 7eqtr4i 2852 1 (𝐸𝑁) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  wnel 3128  {crab 3147  cun 3938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-nel 3129  df-ral 3148  df-rab 3152  df-v 3502  df-un 3945
This theorem is referenced by:  usgrfilem  27042  cusgrsizeinds  27167  vtxdginducedm1  27258
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