Theorem sn0disj 4127

Description: A collection, each of whose elements is empty or a singleton of the index, is disjoint. (Contributed by Jim Kingdon, 8-Oct-2026.)

Hypothesis

Ref	Expression
sn0disj.1	|- ( ( ph /\ x e. A ) -> ( B = (/) \/ B = { x } ) )

Assertion

Ref	Expression
sn0disj	|- ( ph -> Disj x e. A B )

Distinct variable group:   ph, x

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem sn0disj

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 533	. . . . . . . . . 10 |- ( ( ( x e. A /\ y e. B ) /\ B = (/) ) -> y e. B )
2		simpr 110	. . . . . . . . . 10 |- ( ( ( x e. A /\ y e. B ) /\ B = (/) ) -> B = (/) )
3	1, 2	eleqtrd 2317	. . . . . . . . 9 |- ( ( ( x e. A /\ y e. B ) /\ B = (/) ) -> y e. (/) )
4		noel 3525	. . . . . . . . . 10 |- -. y e. (/)
5	4	a1i 9	. . . . . . . . 9 |- ( ( ( x e. A /\ y e. B ) /\ B = (/) ) -> -. y e. (/) )
6	3, 5	pm2.21dd 629	. . . . . . . 8 |- ( ( ( x e. A /\ y e. B ) /\ B = (/) ) -> x = y )
7	6	adantll 480	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ B = (/) ) -> x = y )
8		simplr 533	. . . . . . . . . . 11 |- ( ( ( x e. A /\ y e. B ) /\ B = { x } ) -> y e. B )
9		simpr 110	. . . . . . . . . . 11 |- ( ( ( x e. A /\ y e. B ) /\ B = { x } ) -> B = { x } )
10	8, 9	eleqtrd 2317	. . . . . . . . . 10 |- ( ( ( x e. A /\ y e. B ) /\ B = { x } ) -> y e. { x } )
11		velsn 3726	. . . . . . . . . 10 |- ( y e. { x } <-> y = x )
12	10, 11	sylib 122	. . . . . . . . 9 |- ( ( ( x e. A /\ y e. B ) /\ B = { x } ) -> y = x )
13	12	equcomd 1759	. . . . . . . 8 |- ( ( ( x e. A /\ y e. B ) /\ B = { x } ) -> x = y )
14	13	adantll 480	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ B = { x } ) -> x = y )
15		sn0disj.1	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( B = (/) \/ B = { x } ) )
16	15	adantrr 483	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( B = (/) \/ B = { x } ) )
17	7, 14, 16	mpjaodan 810	. . . . . 6 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> x = y )
18	17	ex 115	. . . . 5 |- ( ph -> ( ( x e. A /\ y e. B ) -> x = y ) )
19	18	alrimiv 1927	. . . 4 |- ( ph -> A. x ( ( x e. A /\ y e. B ) -> x = y ) )
20		mo2icl 3005	. . . 4 |- ( A. x ( ( x e. A /\ y e. B ) -> x = y ) -> E* x ( x e. A /\ y e. B ) )
21	19, 20	syl 14	. . 3 |- ( ph -> E* x ( x e. A /\ y e. B ) )
22	21	alrimiv 1927	. 2 |- ( ph -> A. y E* x ( x e. A /\ y e. B ) )
23		dfdisj2 4108	. 2 |- (Disj x e. A B <-> A. y E* x ( x e. A /\ y e. B ) )
24	22, 23	sylibr 134	1 |- ( ph -> Disj x e. A B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   A.wal 1400   = wceq 1402   E*wmo 2087   e. wcel 2209   (/)c0 3520   {csn 3709  Disj wdisj 4106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rmo 2536  df-v 2823  df-dif 3222  df-nul 3521  df-sn 3715  df-disj 4107

This theorem is used by:  gsum2d2fi  14274