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Theorem iunsuc 4655
 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 4579 . . 3
2 iuneq1 4098 . . 3
31, 2ax-mp 8 . 2
4 iunxun 4164 . 2
5 iunsuc.1 . . . 4
6 iunsuc.2 . . . 4
75, 6iunxsn 4162 . . 3
87uneq2i 3490 . 2
93, 4, 83eqtri 2459 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2948   cun 3310  csn 3806  ciun 4085   csuc 4575 This theorem is referenced by:  pwsdompw  8076 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-un 3317  df-in 3319  df-ss 3326  df-sn 3812  df-iun 4087  df-suc 4579
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