Theorem superuncl 37873

Description: The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem superuncl

Step	Hyp	Ref	Expression
1		superficl.a	. 2 |- A = { z | B C_ z }
2		vex 3203	. . 3 |- x e. _V
3		vex 3203	. . 3 |- y e. _V
4	2, 3	unex 6956	. 2 |- ( x u. y ) e. _V
5		sseq2 3627	. 2 |- ( z = ( x u. y ) -> ( B C_ z <-> B C_ ( x u. y ) ) )
6		sseq2 3627	. 2 |- ( z = x -> ( B C_ z <-> B C_ x ) )
7		sseq2 3627	. 2 |- ( z = y -> ( B C_ z <-> B C_ y ) )
8		ssun3 3778	. . 3 |- ( B C_ x -> B C_ ( x u. y ) )
9	8	adantr 481	. 2 |- ( ( B C_ x /\ B C_ y ) -> B C_ ( x u. y ) )
10	1, 4, 5, 6, 7, 9	cllem0 37871	1 |- A. x e. A A. y e. A ( x u. y ) e. A

Syntax hints:   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   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)