Theorem disjorf 29392

Description: Two ways to say that a collection B ( i ) for i e. A is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypotheses

Ref	Expression
disjorf.1	|- F/_ i A
disjorf.2	|- F/_ j A
disjorf.3	|- ( i = j -> B = C )

Assertion

Ref	Expression
disjorf	|- (Disj i e. A B <-> A. i e. A A. j e. A ( i = j \/ ( B i^i C ) = (/) ) )

Distinct variable groups:   i, j   B, j   C, i

Allowed substitution hints:   A( i, j)   B( i)   C( j)

Proof of Theorem disjorf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 4621	. 2 |- (Disj i e. A B <-> A. x E* i e. A x e. B )
2		ralcom4 3224	. . 3 |- ( A. i e. A A. x A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) <-> A. x A. i e. A A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
3		orcom 402	. . . . . . 7 |- ( ( i = j \/ ( B i^i C ) = (/) ) <-> ( ( B i^i C ) = (/) \/ i = j ) )
4		df-or 385	. . . . . . 7 |- ( ( ( B i^i C ) = (/) \/ i = j ) <-> ( -. ( B i^i C ) = (/) -> i = j ) )
5		neq0 3930	. . . . . . . . . 10 |- ( -. ( B i^i C ) = (/) <-> E. x x e. ( B i^i C ) )
6		elin 3796	. . . . . . . . . . 11 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
7	6	exbii 1774	. . . . . . . . . 10 |- ( E. x x e. ( B i^i C ) <-> E. x ( x e. B /\ x e. C ) )
8	5, 7	bitri 264	. . . . . . . . 9 |- ( -. ( B i^i C ) = (/) <-> E. x ( x e. B /\ x e. C ) )
9	8	imbi1i 339	. . . . . . . 8 |- ( ( -. ( B i^i C ) = (/) -> i = j ) <-> ( E. x ( x e. B /\ x e. C ) -> i = j ) )
10		19.23v 1902	. . . . . . . 8 |- ( A. x ( ( x e. B /\ x e. C ) -> i = j ) <-> ( E. x ( x e. B /\ x e. C ) -> i = j ) )
11	9, 10	bitr4i 267	. . . . . . 7 |- ( ( -. ( B i^i C ) = (/) -> i = j ) <-> A. x ( ( x e. B /\ x e. C ) -> i = j ) )
12	3, 4, 11	3bitri 286	. . . . . 6 |- ( ( i = j \/ ( B i^i C ) = (/) ) <-> A. x ( ( x e. B /\ x e. C ) -> i = j ) )
13	12	ralbii 2980	. . . . 5 |- ( A. j e. A ( i = j \/ ( B i^i C ) = (/) ) <-> A. j e. A A. x ( ( x e. B /\ x e. C ) -> i = j ) )
14		ralcom4 3224	. . . . 5 |- ( A. j e. A A. x ( ( x e. B /\ x e. C ) -> i = j ) <-> A. x A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
15	13, 14	bitri 264	. . . 4 |- ( A. j e. A ( i = j \/ ( B i^i C ) = (/) ) <-> A. x A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
16	15	ralbii 2980	. . 3 |- ( A. i e. A A. j e. A ( i = j \/ ( B i^i C ) = (/) ) <-> A. i e. A A. x A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
17		disjorf.1	. . . . 5 |- F/_ i A
18		disjorf.2	. . . . 5 |- F/_ j A
19		nfv 1843	. . . . 5 |- F/ i x e. C
20		disjorf.3	. . . . . 6 |- ( i = j -> B = C )
21	20	eleq2d 2687	. . . . 5 |- ( i = j -> ( x e. B <-> x e. C ) )
22	17, 18, 19, 21	rmo4f 29337	. . . 4 |- ( E* i e. A x e. B <-> A. i e. A A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
23	22	albii 1747	. . 3 |- ( A. x E* i e. A x e. B <-> A. x A. i e. A A. j e. A ( ( x e. B /\ x e. C ) -> i = j ) )
24	2, 16, 23	3bitr4i 292	. 2 |- ( A. i e. A A. j e. A ( i = j \/ ( B i^i C ) = (/) ) <-> A. x E* i e. A x e. B )
25	1, 24	bitr4i 267	1 |- (Disj i e. A B <-> A. i e. A A. j e. A ( i = j \/ ( B i^i C ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E*wrmo 2915   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-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by: (None)