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Theorem disjss3 4214
 Description: Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Assertion
Ref Expression
disjss3 Disj Disj
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem disjss3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ral 2712 . . . . . . 7
2 simp3r 987 . . . . . . . . . . . 12
3 n0i 3635 . . . . . . . . . . . 12
42, 3syl 16 . . . . . . . . . . 11
5 simp3l 986 . . . . . . . . . . . 12
6 eldif 3332 . . . . . . . . . . . . 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 1632 . . . . . . 7
141, 13syl5bi 210 . . . . . 6
1514imp 420 . . . . 5
16 moim 2329 . . . . 5
1715, 16syl 16 . . . 4
1817alimdv 1632 . . 3
19 dfdisj2 4187 . . 3 Disj
20 dfdisj2 4187 . . 3 Disj
2118, 19, 203imtr4g 263 . 2 Disj Disj
22 disjss1 4191 . . 3 Disj Disj
2322adantr 453 . 2 Disj Disj
2421, 23impbid 185 1 Disj Disj
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360   w3a 937  wal 1550   wceq 1653   wcel 1726  wmo 2284  wral 2707   cdif 3319   wss 3322  c0 3630  Disj wdisj 4185 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 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rmo 2715  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-disj 4186
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