Theorem disjun0 29408

Description: Adding the empty element preserves disjointness. (Contributed by Thierry Arnoux, 30-May-2020.)

Assertion

Ref	Expression
disjun0	|- (Disj x e. A x -> Disj x e. ( A u. { (/) } ) x )

Distinct variable group:   x, A

Proof of Theorem disjun0

Step	Hyp	Ref	Expression
1		snssi 4339	. . . . 5 |- ( (/) e. A -> { (/) } C_ A )
2		ssequn2 3786	. . . . 5 |- ( { (/) } C_ A <-> ( A u. { (/) } ) = A )
3	1, 2	sylib 208	. . . 4 |- ( (/) e. A -> ( A u. { (/) } ) = A )
4	3	disjeq1d 4628	. . 3 |- ( (/) e. A -> (Disj x e. ( A u. { (/) } ) x <-> Disj x e. A x ) )
5	4	biimparc 504	. 2 |- ( (Disj x e. A x /\ (/) e. A ) -> Disj x e. ( A u. { (/) } ) x )
6		simpl 473	. . 3 |- ( (Disj x e. A x /\ -. (/) e. A ) -> Disj x e. A x )
7		in0 3968	. . . 4 |- ( U_ x e. A x i^i (/) ) = (/)
8	7	a1i 11	. . 3 |- ( (Disj x e. A x /\ -. (/) e. A ) -> ( U_ x e. A x i^i (/) ) = (/) )
9		0ex 4790	. . . . 5 |- (/) e. _V
10		id 22	. . . . . 6 |- ( x = (/) -> x = (/) )
11	10	disjunsn 29407	. . . . 5 |- ( ( (/) e. _V /\ -. (/) e. A ) -> (Disj x e. ( A u. { (/) } ) x <-> (Disj x e. A x /\ ( U_ x e. A x i^i (/) ) = (/) ) ) )
12	9, 11	mpan 706	. . . 4 |- ( -. (/) e. A -> (Disj x e. ( A u. { (/) } ) x <-> (Disj x e. A x /\ ( U_ x e. A x i^i (/) ) = (/) ) ) )
13	12	adantl 482	. . 3 |- ( (Disj x e. A x /\ -. (/) e. A ) -> (Disj x e. ( A u. { (/) } ) x <-> (Disj x e. A x /\ ( U_ x e. A x i^i (/) ) = (/) ) ) )
14	6, 8, 13	mpbir2and 957	. 2 |- ( (Disj x e. A x /\ -. (/) e. A ) -> Disj x e. ( A u. { (/) } ) x )
15	5, 14	pm2.61dan 832	1 |- (Disj x e. A x -> Disj x e. ( A u. { (/) } ) x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520  Disj wdisj 4620

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  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-iun 4522  df-disj 4621

This theorem is referenced by:  carsggect  30380