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Theorem elirrv 8130
 Description: The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 8135 and efrirr 4820, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
elirrv

Proof of Theorem elirrv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2537 . . . 4
2 ssnid 3989 . . . 4
31, 2spei 2118 . . 3
4 snex 4641 . . . 4
54zfregcl 8127 . . 3
63, 5ax-mp 5 . 2
7 elsn 3973 . . . . . . 7
8 ax9 1917 . . . . . . . . 9
98equcoms 1872 . . . . . . . 8
109com12 31 . . . . . . 7
117, 10syl5bi 225 . . . . . 6
12 eleq1 2537 . . . . . . . . 9
1312notbid 301 . . . . . . . 8
1413rspccv 3133 . . . . . . 7
152, 14mt2i 122 . . . . . 6
1611, 15nsyli 148 . . . . 5
1716con2d 119 . . . 4
1817ralrimiv 2808 . . 3
19 ralnex 2834 . . 3
2018, 19sylib 201 . 2
216, 20mt2 184 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4  wex 1671   wcel 1904  wral 2756  wrex 2757  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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-reg 8125 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-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962 This theorem is referenced by:  elirr  8131  ruv  8133  dfac2  8579  nd1  9030  nd2  9031  nd3  9032  axunnd  9039  axregndlem1  9045  axregndlem2  9046  axregnd  9047  elpotr  30498  exnel  30520  distel  30521
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