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Theorem elirrv 7195
 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 7200 and efrirr 4267, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
elirrv

Proof of Theorem elirrv
StepHypRef Expression
1 eleq1 2313 . . . 4
2 vex 2730 . . . . 5
32snid 3571 . . . 4
41, 3a4eiv 1998 . . 3
5 snex 4110 . . . 4
65zfregcl 7192 . . 3
74, 6ax-mp 10 . 2
8 elsn 3559 . . . . . . 7
9 ax-14 1626 . . . . . . . . 9
109equcoms 1825 . . . . . . . 8
1110com12 29 . . . . . . 7
128, 11syl5bi 210 . . . . . 6
13 eleq1 2313 . . . . . . . . 9
1413notbid 287 . . . . . . . 8
1514rcla4cv 2818 . . . . . . 7
163, 15mt2i 112 . . . . . 6
1712, 16nsyli 135 . . . . 5
1817con2d 109 . . . 4
1918ralrimiv 2587 . . 3
20 ralnex 2517 . . 3
2119, 20sylib 190 . 2
227, 21mt2 172 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6  wex 1537   wceq 1619   wcel 1621  wral 2509  wrex 2510  csn 3544 This theorem is referenced by:  elirr  7196  ruv  7198  dfac2  7641  nd1  8089  nd2  8090  nd3  8091  axunnd  8098  axregndlem1  8104  axregndlem2  8105  axregnd  8106  elpotr  23305  distel  23328 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-reg 7190 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-un 3083  df-nul 3363  df-sn 3550  df-pr 3551
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