Theorem disjnims 3974

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 2308	. . 3 |- F/_ i B
2		nfcsb1v 3078	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3054	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 3969	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3048	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjnim 3973	. 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 1343   =/= wne 2336   A.wral 2444   [_csb 3045   i^i cin 3115   (/)c0 3409  Disj wdisj 3959

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-in 3122  df-nul 3410  df-disj 3960

This theorem is referenced by:  disji2  3975