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Theorem morex 2719
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1  _V
morex.2
Assertion
Ref Expression
morex
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem morex
StepHypRef Expression
1 df-rex 2306 . . . 4
2 exancom 1496 . . . 4
31, 2bitri 173 . . 3
4 nfmo1 1909 . . . . . 6  F/
5 nfe1 1382 . . . . . 6  F/
64, 5nfan 1454 . . . . 5  F/
7 mopick 1975 . . . . 5
86, 7alrimi 1412 . . . 4
9 morex.1 . . . . 5  _V
10 morex.2 . . . . . 6
11 eleq1 2097 . . . . . 6
1210, 11imbi12d 223 . . . . 5
139, 12spcv 2640 . . . 4
148, 13syl 14 . . 3
153, 14sylan2b 271 . 2
1615ancoms 255 1
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390  wmo 1898  wrex 2301   _Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by: (None)
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