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Theorem dmsnsnsn 5492
 Description: The domain of the singleton of the singleton of a singleton. (Contributed by NM, 15-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnsnsn

Proof of Theorem dmsnsnsn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . . . 8
21opid 4238 . . . . . . 7
3 sneq 4042 . . . . . . . 8
43sneqd 4044 . . . . . . 7
52, 4syl5eq 2510 . . . . . 6
65sneqd 4044 . . . . 5
76dmeqd 5215 . . . 4
87, 3eqeq12d 2479 . . 3
91dmsnop 5488 . . 3
108, 9vtoclg 3167 . 2
11 0ex 4587 . . . . 5
1211snid 4060 . . . 4
13 dmsn0el 5483 . . . 4
1412, 13ax-mp 5 . . 3
15 snprc 4095 . . . . . . 7
1615biimpi 194 . . . . . 6
1716sneqd 4044 . . . . 5
1817sneqd 4044 . . . 4
1918dmeqd 5215 . . 3
2014, 19, 163eqtr4a 2524 . 2
2110, 20pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1395   wcel 1819  cvv 3109  c0 3793  csn 4032  cop 4038   cdm 5008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-dm 5018 This theorem is referenced by: (None)
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