Theorem disjors 4635

Description: Two ways to say that a collection B ( i ) for i e. A is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjors	|- (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 disjors

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ i B
2		nfcsb1v 3549	. . 3 |- F/_ x [_ i / x ]_ B
3		csbeq1a 3542	. . 3 |- ( x = i -> B = [_ i / x ]_ B )
4	1, 2, 3	cbvdisj 4630	. 2 |- (Disj x e. A B <-> Disj i e. A [_ i / x ]_ B )
5		csbeq1 3536	. . 3 |- ( i = j -> [_ i / x ]_ B = [_ j / x ]_ B )
6	5	disjor 4634	. 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	bitri 264	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:   <-> wb 196   \/ wo 383   = wceq 1483   A.wral 2912   [_csb 3533   i^i cin 3573   (/)c0 3915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  disji2  4636  disjprg  4648  disjxiun  4649  disjxiunOLD  4650  disjxun  4651  iundisj2  23317  disji2f  29390  disjpreima  29397  disjxpin  29401  iundisj2f  29403  disjunsn  29407  iundisj2fi  29556  disjxp1  39238  disjinfi  39380