Theorem disjnims 4025

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 2339	. . 3 |- F/_ i B
2		nfcsb1v 3117	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3093	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 4020	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3087	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjnim 4024	. 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 121	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 1364   =/= wne 2367   A.wral 2475   [_csb 3084   i^i cin 3156   (/)c0 3450  Disj wdisj 4010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-in 3163  df-nul 3451  df-disj 4011

This theorem is referenced by:  disji2  4026