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Theorem elirrv 8489
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 8494 and efrirr 5085, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
elirrv ¬ 𝑥𝑥

Proof of Theorem elirrv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4899 . . 3 {𝑥} ∈ V
2 eleq1 2687 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3 vsnid 4200 . . . 4 𝑥 ∈ {𝑥}
42, 3spei 2259 . . 3 𝑦 𝑦 ∈ {𝑥}
5 zfregcl 8484 . . 3 ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
61, 4, 5mp2 9 . 2 𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}
7 velsn 4184 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8 ax9 2001 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
98equcoms 1945 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑥𝑥𝑦))
109com12 32 . . . . . . 7 (𝑥𝑥 → (𝑦 = 𝑥𝑥𝑦))
117, 10syl5bi 232 . . . . . 6 (𝑥𝑥 → (𝑦 ∈ {𝑥} → 𝑥𝑦))
12 eleq1 2687 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
1312notbid 308 . . . . . . . 8 (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
1413rspccv 3301 . . . . . . 7 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥𝑦 → ¬ 𝑥 ∈ {𝑥}))
153, 14mt2i 132 . . . . . 6 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥𝑦)
1611, 15nsyli 155 . . . . 5 (𝑥𝑥 → (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
1716con2d 129 . . . 4 (𝑥𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
1817ralrimiv 2962 . . 3 (𝑥𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
19 ralnex 2989 . . 3 (∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
2018, 19sylib 208 . 2 (𝑥𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
216, 20mt2 191 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  {csn 4168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-reg 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171
This theorem is referenced by:  elirr  8490  ruv  8492  dfac2  8938  nd1  9394  nd2  9395  nd3  9396  axunnd  9403  axregndlem1  9409  axregndlem2  9410  axregnd  9411  elpotr  31660  exnel  31682  distel  31683
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