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Theorem elirrv 7311
 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 7316 and efrirr 4374, 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 2343 . . . 4
2 vex 2791 . . . . 5
32snid 3667 . . . 4
41, 3speiv 1940 . . 3
5 snex 4216 . . . 4
65zfregcl 7308 . . 3
74, 6ax-mp 8 . 2
8 elsn 3655 . . . . . . 7
9 ax-14 1688 . . . . . . . . 9
109equcoms 1651 . . . . . . . 8
1110com12 27 . . . . . . 7
128, 11syl5bi 208 . . . . . 6
13 eleq1 2343 . . . . . . . . 9
1413notbid 285 . . . . . . . 8
1514rspccv 2881 . . . . . . 7
163, 15mt2i 110 . . . . . 6
1712, 16nsyli 133 . . . . 5
1817con2d 107 . . . 4
1918ralrimiv 2625 . . 3
20 ralnex 2553 . . 3
2119, 20sylib 188 . 2
227, 21mt2 170 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4  wex 1528   wceq 1623   wcel 1684  wral 2543  wrex 2544  csn 3640 This theorem is referenced by:  elirr  7312  ruv  7314  dfac2  7757  nd1  8209  nd2  8210  nd3  8211  axunnd  8218  axregndlem1  8224  axregndlem2  8225  axregnd  8226  elpotr  24137  distel  24160 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
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