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Theorem exmidsssnc 4159
 Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4154 but lets you choose any set as the element of the singleton rather than just . It is similar to exmidsssn 4158 but for a particular set rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
Assertion
Ref Expression
exmidsssnc EXMID
Distinct variable groups:   ,   ,

Proof of Theorem exmidsssnc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exmidsssn 4158 . . . 4 EXMID
2 sneq 3567 . . . . . . . 8
32sseq2d 3154 . . . . . . 7
42eqeq2d 2166 . . . . . . . 8
54orbi2d 780 . . . . . . 7
63, 5bibi12d 234 . . . . . 6
76spcgv 2796 . . . . 5
87alimdv 1856 . . . 4
91, 8syl5bi 151 . . 3 EXMID
10 biimp 117 . . . 4
1110alimi 1432 . . 3
129, 11syl6 33 . 2 EXMID
13 ssrab2 3209 . . . . . . . . 9
14 snexg 4140 . . . . . . . . . 10
15 rabexg 4103 . . . . . . . . . 10
16 sseq1 3147 . . . . . . . . . . . 12
17 eqeq1 2161 . . . . . . . . . . . . 13
18 eqeq1 2161 . . . . . . . . . . . . 13
1917, 18orbi12d 783 . . . . . . . . . . . 12
2016, 19imbi12d 233 . . . . . . . . . . 11
2120spcgv 2796 . . . . . . . . . 10
2214, 15, 213syl 17 . . . . . . . . 9
2313, 22mpii 44 . . . . . . . 8
24 rabeq0 3419 . . . . . . . . . . 11
25 snmg 3673 . . . . . . . . . . . 12
26 r19.3rmv 3480 . . . . . . . . . . . 12
2725, 26syl 14 . . . . . . . . . . 11
2824, 27bitr4id 198 . . . . . . . . . 10
2928biimpd 143 . . . . . . . . 9
30 snidg 3585 . . . . . . . . . . . . 13
3130adantr 274 . . . . . . . . . . . 12
32 simpr 109 . . . . . . . . . . . 12
3331, 32eleqtrrd 2234 . . . . . . . . . . 11
34 biidd 171 . . . . . . . . . . . . 13
3534elrab 2864 . . . . . . . . . . . 12
3635simprbi 273 . . . . . . . . . . 11
3733, 36syl 14 . . . . . . . . . 10
3837ex 114 . . . . . . . . 9
3929, 38orim12d 776 . . . . . . . 8
4023, 39syld 45 . . . . . . 7
41 orcom 718 . . . . . . 7
4240, 41syl6ib 160 . . . . . 6
43 df-dc 821 . . . . . 6 DECID
4442, 43syl6ibr 161 . . . . 5 DECID
4544a1dd 48 . . . 4 DECID
4645alrimdv 1853 . . 3 DECID
47 df-exmid 4151 . . 3 EXMID DECID
4846, 47syl6ibr 161 . 2 EXMID
4912, 48impbid 128 1 EXMID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wb 104   wo 698  DECID wdc 820  wal 1330   wceq 1332  wex 1469   wcel 2125  wral 2432  crab 2436  cvv 2709   wss 3098  c0 3390  csn 3556  EXMIDwem 4150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rab 2441  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-exmid 4151 This theorem is referenced by:  exmidunben  12114
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