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Theorem disjss1 3742
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss1 
C_ Disj  C Disj  C
Distinct variable groups:   ,   ,
Allowed substitution hint:    C()

Proof of Theorem disjss1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 2933 . . . . . 6 
C_
21anim1d 319 . . . . 5 
C_  C  C
32alrimiv 1751 . . . 4 
C_  C  C
4 moim 1961 . . . 4  C  C  C  C
53, 4syl 14 . . 3 
C_  C  C
65alimdv 1756 . 2 
C_  C  C
7 dfdisj2 3738 . 2 Disj  C  C
8 dfdisj2 3738 . 2 Disj  C  C
96, 7, 83imtr4g 194 1 
C_ Disj  C Disj  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wcel 1390  wmo 1898    C_ wss 2911  Disj wdisj 3736
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-bndl 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-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-rmo 2308  df-in 2918  df-ss 2925  df-disj 3737
This theorem is referenced by:  disjeq1  3743  disjx0  3754
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