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Theorem disjss3 4242
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 2717 . . . . . . 7
2 simp3r 987 . . . . . . . . . . . 12
3 n0i 3621 . . . . . . . . . . . 12
42, 3syl 16 . . . . . . . . . . 11
5 simp3l 986 . . . . . . . . . . . 12
6 eldif 3319 . . . . . . . . . . . . 13
7 simp2 959 . . . . . . . . . . . . 13
86, 7syl5bir 211 . . . . . . . . . . . 12
95, 8mpand 658 . . . . . . . . . . 11
104, 9mt3d 120 . . . . . . . . . 10
1110, 2jca 520 . . . . . . . . 9
12113exp 1153 . . . . . . . 8
1312alimdv 1633 . . . . . . 7
141, 13syl5bi 210 . . . . . 6
1514imp 420 . . . . 5
16 moim 2334 . . . . 5
1715, 16syl 16 . . . 4
1817alimdv 1633 . . 3
19 dfdisj2 4215 . . 3
20 dfdisj2 4215 . . 3
2118, 19, 203imtr4g 263 . 2
22 disjss1 4219 . . 3
2322adantr 453 . 2
2421, 23impbid 185 1
Colors of variables: wff set class
Syntax hints:  -.wn 3  ->wi 4  <->wb 178  /\wa 360  /\w3a 937  A.wal 1550  =wceq 1654  e.wcel 1728  E*wmo 2289  A.wral 2712  \cdif 3306  C_wss 3309   c0 3616  Disj_wdisj 4213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2717  df-rmo 2720  df-v 2967  df-dif 3312  df-in 3316  df-ss 3323  df-nul 3617  df-disj 4214
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