Theorem disjnims 4036

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 2348	. . 3 |- F/_ i B
2		nfcsb1v 3126	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3102	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 4031	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3096	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjnim 4035	. 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 1373   =/= wne 2376   A.wral 2484   [_csb 3093   i^i cin 3165   (/)c0 3460  Disj wdisj 4021

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-in 3172  df-nul 3461  df-disj 4022

This theorem is referenced by:  disji2  4037