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Theorem brprcneu 5872
 Description: If is a proper class, then there is no unique binary relationship with as the first element. (Contributed by Scott Fenton, 7-Oct-2017.)
Assertion
Ref Expression
brprcneu
Distinct variable groups:   ,   ,

Proof of Theorem brprcneu
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dtru 4592 . . . . . . . . 9
2 exnal 1707 . . . . . . . . . 10
3 equcom 1870 . . . . . . . . . . 11
43albii 1699 . . . . . . . . . 10
52, 4xchbinx 317 . . . . . . . . 9
61, 5mpbir 214 . . . . . . . 8
76jctr 551 . . . . . . 7
8 19.42v 1842 . . . . . . 7
97, 8sylibr 217 . . . . . 6
10 opprc1 4181 . . . . . . . 8
1110eleq1d 2533 . . . . . . 7
12 opprc1 4181 . . . . . . . . . . . 12
1312eleq1d 2533 . . . . . . . . . . 11
1411, 13anbi12d 725 . . . . . . . . . 10
15 anidm 656 . . . . . . . . . 10
1614, 15syl6bb 269 . . . . . . . . 9
1716anbi1d 719 . . . . . . . 8
1817exbidv 1776 . . . . . . 7
1911, 18imbi12d 327 . . . . . 6
209, 19mpbiri 241 . . . . 5
21 df-br 4396 . . . . 5
22 df-br 4396 . . . . . . . 8
2321, 22anbi12i 711 . . . . . . 7
2423anbi1i 709 . . . . . 6
2524exbii 1726 . . . . 5
2620, 21, 253imtr4g 278 . . . 4
2726eximdv 1772 . . 3
28 exnal 1707 . . . 4
29 exanali 1729 . . . . 5
3029exbii 1726 . . . 4
31 breq2 4399 . . . . . 6
3231mo4 2366 . . . . 5
3332notbii 303 . . . 4
3428, 30, 333bitr4ri 286 . . 3
3527, 34syl6ibr 235 . 2
36 eu5 2345 . . . 4
3736notbii 303 . . 3
38 imnan 429 . . 3
3937, 38bitr4i 260 . 2
4035, 39sylibr 217 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 376  wal 1450  wex 1671   wcel 1904  weu 2319  wmo 2320  cvv 3031  c0 3722  cop 3965   class class class wbr 4395 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527  ax-pow 4579 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396 This theorem is referenced by:  fvprc  5873
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