Theorem sssymdifcl 37877

Description: The class of all subsets of a class is closed under symmetric difference. (Contributed by Richard Penner, 3-Jan-2020.)

Hypothesis

Ref	Expression
ssficl.a	|- A = { z | z C_ B }

Assertion

Ref	Expression
sssymdifcl	|- A. x e. A A. y e. A ( ( x \ y ) u. ( y \ x ) ) e. A

Distinct variable groups:   x, y, z   y, A   z, B

Allowed substitution hints:   A( x, z)   B( x, y)

Proof of Theorem sssymdifcl

Step	Hyp	Ref	Expression
1		ssficl.a	. 2 |- A = { z | z C_ B }
2		vex 3203	. . . 4 |- x e. _V
3		difexg 4808	. . . 4 |- ( x e. _V -> ( x \ y ) e. _V )
4	2, 3	ax-mp 5	. . 3 |- ( x \ y ) e. _V
5		vex 3203	. . . 4 |- y e. _V
6		difexg 4808	. . . 4 |- ( y e. _V -> ( y \ x ) e. _V )
7	5, 6	ax-mp 5	. . 3 |- ( y \ x ) e. _V
8	4, 7	unex 6956	. 2 |- ( ( x \ y ) u. ( y \ x ) ) e. _V
9		sseq1 3626	. 2 |- ( z = ( ( x \ y ) u. ( y \ x ) ) -> ( z C_ B <-> ( ( x \ y ) u. ( y \ x ) ) C_ B ) )
10		sseq1 3626	. 2 |- ( z = x -> ( z C_ B <-> x C_ B ) )
11		sseq1 3626	. 2 |- ( z = y -> ( z C_ B <-> y C_ B ) )
12		ssdifss 3741	. . 3 |- ( x C_ B -> ( x \ y ) C_ B )
13		ssdifss 3741	. . 3 |- ( y C_ B -> ( y \ x ) C_ B )
14		unss 3787	. . . 4 |- ( ( ( x \ y ) C_ B /\ ( y \ x ) C_ B ) <-> ( ( x \ y ) u. ( y \ x ) ) C_ B )
15	14	biimpi 206	. . 3 |- ( ( ( x \ y ) C_ B /\ ( y \ x ) C_ B ) -> ( ( x \ y ) u. ( y \ x ) ) C_ B )
16	12, 13, 15	syl2an 494	. 2 |- ( ( x C_ B /\ y C_ B ) -> ( ( x \ y ) u. ( y \ x ) ) C_ B )
17	1, 8, 9, 10, 11, 16	cllem0 37871	1 |- A. x e. A A. y e. A ( ( x \ y ) u. ( y \ x ) ) e. A

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)