Theorem disjnims 3929

Description: If a collection B ( i ) for i e. A is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)

Assertion

Ref	Expression
disjnims	|- (Disj x e. A B -> A. i e. A A. j e. A ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) )

Distinct variable groups:   i, j, x, A   B, i, j

Allowed substitution hint:   B( x)

Proof of Theorem disjnims

Step	Hyp	Ref	Expression
1		nfcv 2282	. . 3 |- F/_ i B
2		nfcsb1v 3040	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3016	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 3924	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3010	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjnim 3928	. 2 |- (Disj i e. A [_ i / x ]_ B -> A. i e. A A. j e. A ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) )
7	4, 6	sylbi 120	1 |- (Disj x e. A B -> A. i e. A A. j e. A ( i =/= j -> ( [_ i / x ]_ B i^i [_ j / x ]_ B ) = (/) ) )

Syntax hints:   -> wi 4   = wceq 1332   =/= wne 2309   A.wral 2417   [_csb 3007   i^i cin 3075   (/)c0 3368  Disj wdisj 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-in 3082  df-nul 3369  df-disj 3915

This theorem is referenced by:  disji2  3930