Theorem iinxsng 4407

Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Hypothesis

Ref	Expression
iinxsng.1

Assertion

Ref	Expression
iinxsng

Distinct variable groups:   ,   ,

Proof of Theorem iinxsng

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 4333	. 2
2		iinxsng.1	. . . . 5
3	2	eleq2d 2527	. . . 4
4	3	ralsng 4064	. . 3
5	4	abbi1dv 2595	. 2
6	1, 5	syl5eq 2510	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  {csn 4029  |^|_ciin 4331

This theorem is referenced by:  polatN  35655

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-ral 2812  df-v 3111  df-sbc 3328  df-sn 4030  df-iin 4333