Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunsuc Structured version   Visualization version   Unicode version

Theorem iunsuc 5508
 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:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iunsuc
StepHypRef Expression
1 df-suc 5432 . . 3
2 iuneq1 4295 . . 3
31, 2ax-mp 5 . 2
4 iunxun 4366 . 2
5 iunsuc.1 . . . 4
6 iunsuc.2 . . . 4
75, 6iunxsn 4364 . . 3
87uneq2i 3587 . 2
93, 4, 83eqtri 2479 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1446   wcel 1889  cvv 3047   cun 3404  csn 3970  ciun 4281   csuc 5428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-v 3049  df-sbc 3270  df-un 3411  df-in 3413  df-ss 3420  df-sn 3971  df-iun 4283  df-suc 5432 This theorem is referenced by:  pwsdompw  8639
 Copyright terms: Public domain W3C validator