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Theorem elirrv 9634
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 9643 and efrirr 5669, 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 vsnex 5440 . . 3 {𝑥} ∈ V
2 eleq1w 2822 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3 vsnid 4668 . . . 4 𝑥 ∈ {𝑥}
42, 3speivw 1971 . . 3 𝑦 𝑦 ∈ {𝑥}
5 zfregcl 9632 . . 3 ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
61, 4, 5mp2 9 . 2 𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}
7 velsn 4647 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8 ax9 2120 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
98equcoms 2017 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑥𝑥𝑦))
109com12 32 . . . . . . 7 (𝑥𝑥 → (𝑦 = 𝑥𝑥𝑦))
117, 10biimtrid 242 . . . . . 6 (𝑥𝑥 → (𝑦 ∈ {𝑥} → 𝑥𝑦))
12 eleq1w 2822 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
1312notbid 318 . . . . . . . 8 (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
1413rspccv 3619 . . . . . . 7 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥𝑦 → ¬ 𝑥 ∈ {𝑥}))
153, 14mt2i 137 . . . . . 6 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥𝑦)
1611, 15nsyli 157 . . . . 5 (𝑥𝑥 → (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
1716con2d 134 . . . 4 (𝑥𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
1817ralrimiv 3143 . . 3 (𝑥𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
19 ralnex 3070 . . 3 (∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
2018, 19sylib 218 . 2 (𝑥𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
216, 20mt2 200 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wex 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by:  elirr  9635  nd1  10625  nd2  10626  nd3  10627  axunnd  10634  axregndlem1  10640  axregndlem2  10641  axregnd  10642  axnulg  35085  elpotr  35763  exnel  35784  distel  35785  ruvALT  42656  onsupmaxb  43228  ralndv1  47055
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