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Theorem eqoreldif 4004
 Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.)
Assertion
Ref Expression
eqoreldif

Proof of Theorem eqoreldif
StepHypRef Expression
1 orc 392 . . . . 5
21a1d 25 . . . 4
3 simprr 774 . . . . . . 7
4 elsni 3985 . . . . . . . . . 10
54a1i 11 . . . . . . . . 9
65con3d 140 . . . . . . . 8
76impcom 437 . . . . . . 7
83, 7eldifd 3401 . . . . . 6
98olcd 400 . . . . 5
109ex 441 . . . 4
112, 10pm2.61i 169 . . 3
1211ex 441 . 2
13 eleq1 2537 . . . . 5
1413biimprd 231 . . . 4
15 eldifi 3544 . . . . 5
1615a1d 25 . . . 4
1714, 16jaoi 386 . . 3
1817com12 31 . 2
1912, 18impbid 195 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 189   wo 375   wa 376   wceq 1452   wcel 1904   cdif 3387  csn 3959 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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-dif 3393  df-sn 3960 This theorem is referenced by:  lcmfunsnlem2  14692
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