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Theorem elirrv 8387
 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 8392 and efrirr 5019, 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 4835 . . 3 {𝑥} ∈ V
2 eleq1 2676 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3 vsnid 4156 . . . 4 𝑥 ∈ {𝑥}
42, 3spei 2249 . . 3 𝑦 𝑦 ∈ {𝑥}
5 zfregcl 8382 . . 3 ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
61, 4, 5mp2 9 . 2 𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}
7 velsn 4141 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8 ax9 1990 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
98equcoms 1934 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑥𝑥𝑦))
109com12 32 . . . . . . 7 (𝑥𝑥 → (𝑦 = 𝑥𝑥𝑦))
117, 10syl5bi 231 . . . . . 6 (𝑥𝑥 → (𝑦 ∈ {𝑥} → 𝑥𝑦))
12 eleq1 2676 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
1312notbid 307 . . . . . . . 8 (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
1413rspccv 3279 . . . . . . 7 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥𝑦 → ¬ 𝑥 ∈ {𝑥}))
153, 14mt2i 131 . . . . . 6 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥𝑦)
1611, 15nsyli 154 . . . . 5 (𝑥𝑥 → (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
1716con2d 128 . . . 4 (𝑥𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
1817ralrimiv 2948 . . 3 (𝑥𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
19 ralnex 2975 . . 3 (∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
2018, 19sylib 207 . 2 (𝑥𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
216, 20mt2 190 1 ¬ 𝑥𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128 This theorem is referenced by:  elirr  8388  ruv  8390  dfac2  8836  nd1  9288  nd2  9289  nd3  9290  axunnd  9297  axregndlem1  9303  axregndlem2  9304  axregnd  9305  elpotr  30930  exnel  30952  distel  30953
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