Theorem elnelun 3964

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 = { s e. A | B e. C }
elneldisj.n	|- N = { s e. A | B e/ C }

Assertion

Ref	Expression
elnelun	|- ( E u. N ) = A

Distinct variable group:   A, s

Allowed substitution hints:   B( s)   C( s)   E( s)   N( s)

Proof of Theorem elnelun

Step	Hyp	Ref	Expression
1		elneldisj.e	. . 3 |- E = { s e. A | B e. C }
2		elneldisj.n	. . . 4 |- N = { s e. A | B e/ C }
3		df-nel 2898	. . . . . 6 |- ( B e/ C <-> -. B e. C )
4	3	a1i 11	. . . . 5 |- ( s e. A -> ( B e/ C <-> -. B e. C ) )
5	4	rabbiia 3185	. . . 4 |- { s e. A | B e/ C } = { s e. A | -. B e. C }
6	2, 5	eqtri 2644	. . 3 |- N = { s e. A | -. B e. C }
7	1, 6	uneq12i 3765	. 2 |- ( E u. N ) = ( { s e. A | B e. C } u. { s e. A | -. B e. C } )
8		rabxm 3961	. 2 |- A = ( { s e. A | B e. C } u. { s e. A | -. B e. C } )
9	7, 8	eqtr4i 2647	1 |- ( E u. N ) = A

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   u. cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rab 2921  df-v 3202  df-un 3579

This theorem is referenced by:  usgrfilem  26219  cusgrsizeinds  26348  vtxdginducedm1  26439