Theorem iunin1 4395

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4383 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
iunin1

Distinct variable group:   ,

Proof of Theorem iunin1

Step	Hyp	Ref	Expression
1		iunin2 4394	. 2
2		incom 3690	. . . 4
3	2	a1i 11	. . 3
4	3	iuneq2i 4349	. 2
5		incom 3690	. 2
6	1, 4, 5	3eqtr4i 2496	1

Syntax hints:  =wceq 1395  e.wcel 1818  i^icin 3474  U_ciun 4330

This theorem is referenced by:  2iunin  4398  tgrest  19660  metnrmlem3  21365  limciun  22298  disjunsn  27453  measinblem  28191  sstotbnd2  30270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3482  df-ss 3489  df-iun 4332