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Theorem elirrv 9044
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 9052 and efrirr 5500, 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 5297 . . 3 {𝑥} ∈ V
2 eleq1w 2872 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3 vsnid 4562 . . . 4 𝑥 ∈ {𝑥}
42, 3speivw 1977 . . 3 𝑦 𝑦 ∈ {𝑥}
5 zfregcl 9042 . . 3 ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
61, 4, 5mp2 9 . 2 𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}
7 velsn 4541 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8 ax9 2125 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
98equcoms 2027 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑥𝑥𝑦))
109com12 32 . . . . . . 7 (𝑥𝑥 → (𝑦 = 𝑥𝑥𝑦))
117, 10syl5bi 245 . . . . . 6 (𝑥𝑥 → (𝑦 ∈ {𝑥} → 𝑥𝑦))
12 eleq1w 2872 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
1312notbid 321 . . . . . . . 8 (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
1413rspccv 3568 . . . . . . 7 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥𝑦 → ¬ 𝑥 ∈ {𝑥}))
153, 14mt2i 139 . . . . . 6 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥𝑦)
1611, 15nsyli 160 . . . . 5 (𝑥𝑥 → (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
1716con2d 136 . . . 4 (𝑥𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
1817ralrimiv 3148 . . 3 (𝑥𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
19 ralnex 3199 . . 3 (∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
2018, 19sylib 221 . 2 (𝑥𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
216, 20mt2 203 1 ¬ 𝑥𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wex 1781  wcel 2111  wral 3106  wrex 3107  Vcvv 3441  {csn 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-reg 9040
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528
This theorem is referenced by:  elirr  9045  ruv  9050  nd1  9998  nd2  9999  nd3  10000  axunnd  10007  axregndlem1  10013  axregndlem2  10014  axregnd  10015  elpotr  33136  exnel  33157  distel  33158  ralndv1  43656
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