Theorem iunxsng 4409

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Hypothesis

Ref	Expression
iunxsng.1

Assertion

Ref	Expression
iunxsng

Distinct variable groups:   ,   ,

Proof of Theorem iunxsng

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4335	. . 3
2		iunxsng.1	. . . . 5
3	2	eleq2d 2527	. . . 4
4	3	rexsng 4065	. . 3
5	1, 4	syl5bb 257	. 2
6	5	eqrdv 2454	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  E.wrex 2808  {csn 4029  U_ciun 4330

This theorem is referenced by:  iunxsn  4410  iunxprg  32302

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-3an 975  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-sbc 3328  df-sn 4030  df-iun 4332