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Theorem disjss3 4203
Description: Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Assertion
Ref Expression
disjss3
Distinct variable groups:   ,   ,
Allowed substitution hint:   ( )

Proof of Theorem disjss3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ral 2702 . . . . . . 7
2 simp3r 986 . . . . . . . . . . . 12
3 n0i 3625 . . . . . . . . . . . 12
42, 3syl 16 . . . . . . . . . . 11
5 simp3l 985 . . . . . . . . . . . 12
6 eldif 3322 . . . . . . . . . . . . 13
7 simp2 958 . . . . . . . . . . . . 13
86, 7syl5bir 210 . . . . . . . . . . . 12
95, 8mpand 657 . . . . . . . . . . 11
104, 9mt3d 119 . . . . . . . . . 10
1110, 2jca 519 . . . . . . . . 9
12113exp 1152 . . . . . . . 8
1312alimdv 1631 . . . . . . 7
141, 13syl5bi 209 . . . . . 6
1514imp 419 . . . . 5
16 moim 2326 . . . . 5
1715, 16syl 16 . . . 4
1817alimdv 1631 . . 3
19 dfdisj2 4176 . . 3
20 dfdisj2 4176 . . 3
2118, 19, 203imtr4g 262 . 2
22 disjss1 4180 . . 3
2322adantr 452 . 2
2421, 23impbid 184 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  ->wi 4  <->wb 177  /\wa 359  /\w3a 936  A.wal 1549  =wceq 1652  e.wcel 1725  E*wmo 2281  A.wral 2697  \cdif 3309  C_wss 3312   c0 3620  Disj_wdisj 4174
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rmo 2705  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-disj 4175
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