Theorem disjiunb 4642

Description: Two ways to say that a collection of index unions C ( i , x ) for i e. A and x e. B is disjoint. (Contributed by AV, 9-Jan-2022.)

Hypotheses

Ref	Expression
disjiunb.1	|- ( i = j -> B = D )
disjiunb.2	|- ( i = j -> C = E )

Assertion

Ref	Expression
disjiunb	|- (Disj i e. A U_ x e. B C <-> A. i e. A A. j e. A ( i = j \/ ( U_ x e. B C i^i U_ x e. D E ) = (/) ) )

Distinct variable groups:   A, i, j   B, j, x   C, j   i, E   D, i, x

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

Proof of Theorem disjiunb

Step	Hyp	Ref	Expression
1		disjiunb.1	. . 3 |- ( i = j -> B = D )
2		disjiunb.2	. . 3 |- ( i = j -> C = E )
3	1, 2	iuneq12d 4546	. 2 |- ( i = j -> U_ x e. B C = U_ x e. D E )
4	3	disjor 4634	1 |- (Disj i e. A U_ x e. B C <-> A. i e. A A. j e. A ( i = j \/ ( U_ x e. B C i^i U_ x e. D E ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   A.wral 2912   i^i cin 3573   (/)c0 3915   U_ciun 4520  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-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iun 4522  df-disj 4621

This theorem is referenced by:  disjiund  4643