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Theorem ssuni 3593
Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni  C_  C  C_  U. C

Proof of Theorem ssuni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . . . . 7
21imbi1d 220 . . . . . 6  U. C  U. C
3 elunii 3576 . . . . . . 7  C  U. C
43expcom 109 . . . . . 6  C  U. C
52, 4vtoclga 2613 . . . . 5  C  U. C
65imim2d 48 . . . 4  C  U. C
76alimdv 1756 . . 3  C  U. C
8 dfss2 2928 . . 3 
C_
9 dfss2 2928 . . 3 
C_  U. C  U. C
107, 8, 93imtr4g 194 . 2  C  C_  C_ 
U. C
1110impcom 116 1  C_  C  C_  U. C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242   wcel 1390    C_ wss 2911   U.cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572
This theorem is referenced by:  elssuni  3599  uniss2  3602  ssorduni  4179
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