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Theorem elirrv 8750
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 8758 and efrirr 5305, 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 5111 . . 3 {𝑥} ∈ V
2 eleq1w 2879 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3 vsnid 4414 . . . 4 𝑥 ∈ {𝑥}
42, 3spei 2437 . . 3 𝑦 𝑦 ∈ {𝑥}
5 zfregcl 8748 . . 3 ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
61, 4, 5mp2 9 . 2 𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}
7 velsn 4397 . . . . . . 7 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8 ax9 2170 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥𝑥𝑥𝑦))
98equcoms 2117 . . . . . . . 8 (𝑦 = 𝑥 → (𝑥𝑥𝑥𝑦))
109com12 32 . . . . . . 7 (𝑥𝑥 → (𝑦 = 𝑥𝑥𝑦))
117, 10syl5bi 233 . . . . . 6 (𝑥𝑥 → (𝑦 ∈ {𝑥} → 𝑥𝑦))
12 eleq1w 2879 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
1312notbid 309 . . . . . . . 8 (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
1413rspccv 3510 . . . . . . 7 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥𝑦 → ¬ 𝑥 ∈ {𝑥}))
153, 14mt2i 134 . . . . . 6 (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥𝑦)
1611, 15nsyli 156 . . . . 5 (𝑥𝑥 → (∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
1716con2d 131 . . . 4 (𝑥𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥}))
1817ralrimiv 3164 . . 3 (𝑥𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
19 ralnex 3191 . . 3 (∀𝑦 ∈ {𝑥} ¬ ∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧𝑦 ¬ 𝑧 ∈ {𝑥})
2018, 19sylib 209 . 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 1859  wcel 2157  wral 3107  wrex 3108  Vcvv 3402  {csn 4381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109  ax-reg 8746
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-dif 3783  df-un 3785  df-nul 4128  df-sn 4382  df-pr 4384
This theorem is referenced by:  elirr  8751  ruv  8756  dfac2OLD  9248  nd1  9704  nd2  9705  nd3  9706  axunnd  9713  axregndlem1  9719  axregndlem2  9720  axregnd  9721  elpotr  32028  exnel  32050  distel  32051
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