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Theorem disjssun 3282
Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjssun  i^i  (/)  C_  u.  C  C_  C

Proof of Theorem disjssun
StepHypRef Expression
1 indi 3181 . . . . 5  i^i  u.  C  i^i  u.  i^i  C
21equncomi 3086 . . . 4  i^i  u.  C  i^i  C  u.  i^i
3 uneq2 3088 . . . . 5  i^i  (/)  i^i  C  u.  i^i  i^i  C  u.  (/)
4 un0 3248 . . . . 5  i^i  C  u.  (/)  i^i  C
53, 4syl6eq 2088 . . . 4  i^i  (/)  i^i  C  u.  i^i  i^i  C
62, 5syl5eq 2084 . . 3  i^i  (/)  i^i  u.  C  i^i  C
76eqeq1d 2048 . 2  i^i  (/)  i^i  u.  C  i^i  C
8 df-ss 2928 . 2 
C_  u.  C  i^i  u.  C
9 df-ss 2928 . 2 
C_  C  i^i  C
107, 8, 93bitr4g 212 1  i^i  (/)  C_  u.  C  C_  C
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1243    u. cun 2912    i^i cin 2913    C_ wss 2914   (/)c0 3221
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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222
This theorem is referenced by: (None)
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