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Theorem undifexmid 4154
 Description: Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3474 and undifdcss 6864 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
Hypothesis
Ref Expression
undifexmid.1
Assertion
Ref Expression
undifexmid
Distinct variable group:   ,,

Proof of Theorem undifexmid
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 0ex 4091 . . . . 5
21snid 3591 . . . 4
3 ssrab2 3213 . . . . 5
4 p0ex 4149 . . . . . . 7
54rabex 4108 . . . . . 6
6 sseq12 3153 . . . . . . 7
7 simpl 108 . . . . . . . . 9
8 simpr 109 . . . . . . . . . 10
98, 7difeq12d 3226 . . . . . . . . 9
107, 9uneq12d 3262 . . . . . . . 8
1110, 8eqeq12d 2172 . . . . . . 7
126, 11bibi12d 234 . . . . . 6
13 undifexmid.1 . . . . . 6
145, 4, 12, 13vtocl2 2767 . . . . 5
153, 14mpbi 144 . . . 4
162, 15eleqtrri 2233 . . 3
17 elun 3248 . . 3
1816, 17mpbi 144 . 2
19 biidd 171 . . . . . 6
2019elrab3 2869 . . . . 5
212, 20ax-mp 5 . . . 4
2221biimpi 119 . . 3
23 eldifn 3230 . . . 4
2423, 21sylnib 666 . . 3
2522, 24orim12i 749 . 2
2618, 25ax-mp 5 1
 Colors of variables: wff set class Syntax hints:   wn 3   wa 103   wb 104   wo 698   wceq 1335   wcel 2128  crab 2439   cdif 3099   cun 3100   wss 3102  c0 3394  csn 3560 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566 This theorem is referenced by: (None)
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