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Theorem ssequn1 3086
Description: A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssequn1 
C_  u.

Proof of Theorem ssequn1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bicom 128 . . . 4
2 pm4.72 724 . . . 4
3 elun 3057 . . . . 5  u.
43bibi1i 217 . . . 4  u.
51, 2, 43bitr4i 201 . . 3  u.
65albii 1335 . 2  u.
7 dfss2 2907 . 2 
C_
8 dfcleq 2012 . 2  u.  u.
96, 7, 83bitr4i 201 1 
C_  u.
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wo 616  wal 1224   wceq 1226   wcel 1370    u. cun 2888    C_ wss 2890
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904
This theorem is referenced by:  ssequn2  3089  uniop  3962  pwssunim  3991  unisuc  4095  unisucg  4096  rdgisucinc  5888  oasuc  5955  omsuc  5962
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