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Theorem ndmaovcl 28035
 Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6225 where it is required that the domain contains the empty set ( ). (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
ndmaov.1
ndmaovcl.2 (())
ndmaovcl.3 (())
Assertion
Ref Expression
ndmaovcl (())

Proof of Theorem ndmaovcl
StepHypRef Expression
1 ndmaovcl.2 . 2 (())
2 opelxp 4901 . . 3
3 ndmaov.1 . . . . . 6
43eqcomi 2440 . . . . 5
54eleq2i 2500 . . . 4
6 ndmaovcl.3 . . . . 5 (())
7 ndmaov 28015 . . . . 5 (())
8 eleq1 2496 . . . . . . 7 (()) (())
98biimpd 199 . . . . . 6 (()) (())
10 vprc 4334 . . . . . . 7
1110pm2.21i 125 . . . . . 6 (())
129, 11syl6com 33 . . . . 5 (()) (()) (())
136, 7, 12mpsyl 61 . . . 4 (())
145, 13sylnbi 298 . . 3 (())
152, 14sylnbir 299 . 2 (())
161, 15pm2.61i 158 1 (())
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2949  cop 3810   cxp 4869   cdm 4871   ((caov 27941 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-opab 4260  df-xp 4877  df-fv 5455  df-dfat 27942  df-afv 27943  df-aov 27944
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