Theorem disjnims 3921

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 2281	. . 3 |- F/_ i B
2		nfcsb1v 3035	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3012	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 3916	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3006	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjnim 3920	. 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 1331   =/= wne 2308   A.wral 2416   [_csb 3003   i^i cin 3070   (/)c0 3363  Disj wdisj 3906

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-in 3077  df-nul 3364  df-disj 3907

This theorem is referenced by:  disji2  3922