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Theorem eu2ndop1stv 27947
 Description: If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
Assertion
Ref Expression
eu2ndop1stv
Distinct variable groups:   ,   ,

Proof of Theorem eu2ndop1stv
StepHypRef Expression
1 euex 2303 . 2
2 nfeu1 2290 . . . 4
3 nfcv 2571 . . . . 5
43nfel1 2581 . . . 4
52, 4nfim 1832 . . 3
6 opprc1 3998 . . . . . . . . 9
76eleq1d 2501 . . . . . . . 8
8 ax-17 1626 . . . . . . . . 9
9 alneu 27946 . . . . . . . . 9
108, 9syl 16 . . . . . . . 8
117, 10syl6bi 220 . . . . . . 7
1211impcom 420 . . . . . 6
137eubidv 2288 . . . . . . . 8
1413notbid 286 . . . . . . 7
1514adantl 453 . . . . . 6
1612, 15mpbird 224 . . . . 5
1716ex 424 . . . 4
1817con4d 99 . . 3
195, 18exlimi 1821 . 2
201, 19mpcom 34 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549  wex 1550   wcel 1725  weu 2280  cvv 2948  c0 3620  cop 3809 This theorem is referenced by:  afveu  27984  tz6.12-afv  28004 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 2416  ax-nul 4330  ax-pow 4369 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-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-op 3815
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