Theorem iunsuc 4965

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
iunsuc.1
iunsuc.2

Assertion

Ref	Expression
iunsuc

Distinct variable groups:   ,   ,

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 4889	. . 3
2		iuneq1 4344	. . 3
3	1, 2	ax-mp 5	. 2
4		iunxun 4412	. 2
5		iunsuc.1	. . . 4
6		iunsuc.2	. . . 4
7	5, 6	iunxsn 4410	. . 3
8	7	uneq2i 3654	. 2
9	3, 4, 8	3eqtri 2490	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  cvv 3109  u.cun 3473  {csn 4029  U_ciun 4330  succsuc 4885

This theorem is referenced by:  pwsdompw  8605

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-or 370  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-un 3480  df-in 3482  df-ss 3489  df-sn 4030  df-iun 4332  df-suc 4889