Theorem disjord 4641

Description: Conditions for a collection of sets A ( a ) for a e. V to be disjoint. (Contributed by AV, 9-Jan-2022.)

Hypotheses

Ref	Expression
disjord.1	|- ( a = b -> A = B )
disjord.2	|- ( ( ph /\ x e. A /\ x e. B ) -> a = b )

Assertion

Ref	Expression
disjord	|- ( ph -> Disj a e. V A )

Distinct variable groups:   A, b, x   B, a, x   V, a, b, x   ph, a, b, x

Allowed substitution hints:   A( a)   B( b)

Proof of Theorem disjord

Step	Hyp	Ref	Expression
1		orc 400	. . . . . 6 |- ( a = b -> ( a = b \/ ( A i^i B ) = (/) ) )
2	1	a1d 25	. . . . 5 |- ( a = b -> ( ph -> ( a = b \/ ( A i^i B ) = (/) ) ) )
3		disjord.2	. . . . . . . . . . . 12 |- ( ( ph /\ x e. A /\ x e. B ) -> a = b )
4	3	3expia 1267	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> ( x e. B -> a = b ) )
5	4	con3d 148	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( -. a = b -> -. x e. B ) )
6	5	impancom 456	. . . . . . . . 9 |- ( ( ph /\ -. a = b ) -> ( x e. A -> -. x e. B ) )
7	6	ralrimiv 2965	. . . . . . . 8 |- ( ( ph /\ -. a = b ) -> A. x e. A -. x e. B )
8		disj 4017	. . . . . . . 8 |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B )
9	7, 8	sylibr 224	. . . . . . 7 |- ( ( ph /\ -. a = b ) -> ( A i^i B ) = (/) )
10	9	olcd 408	. . . . . 6 |- ( ( ph /\ -. a = b ) -> ( a = b \/ ( A i^i B ) = (/) ) )
11	10	expcom 451	. . . . 5 |- ( -. a = b -> ( ph -> ( a = b \/ ( A i^i B ) = (/) ) ) )
12	2, 11	pm2.61i 176	. . . 4 |- ( ph -> ( a = b \/ ( A i^i B ) = (/) ) )
13	12	adantr 481	. . 3 |- ( ( ph /\ ( a e. V /\ b e. V ) ) -> ( a = b \/ ( A i^i B ) = (/) ) )
14	13	ralrimivva 2971	. 2 |- ( ph -> A. a e. V A. b e. V ( a = b \/ ( A i^i B ) = (/) ) )
15		disjord.1	. . 3 |- ( a = b -> A = B )
16	15	disjor 4634	. 2 |- (Disj a e. V A <-> A. a e. V A. b e. V ( a = b \/ ( A i^i B ) = (/) ) )
17	14, 16	sylibr 224	1 |- ( ph -> Disj a e. V A )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   (/)c0 3915  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

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-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  2wspdisj  26855