Theorem elirrv 9355

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 9363 and efrirr 5570, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)

Assertion

Ref	Expression
elirrv	⊢ ¬ 𝑥 ∈ 𝑥

Proof of Theorem elirrv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5354	. . 3 ⊢ {𝑥} ∈ V
2		eleq1w 2821	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3		vsnid 4598	. . . 4 ⊢ 𝑥 ∈ {𝑥}
4	2, 3	speivw 1977	. . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥}
5		zfregcl 9353	. . 3 ⊢ ({𝑥} ∈ V → (∃𝑦 𝑦 ∈ {𝑥} → ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}))
6	1, 4, 5	mp2 9	. 2 ⊢ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}
7		velsn 4577	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8		ax9 2120	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
9	8	equcoms 2023	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
10	9	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦))
11	7, 10	syl5bi 241	. . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦))
12		eleq1w 2821	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
13	12	notbid 318	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
14	13	rspccv 3558	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥}))
15	3, 14	mt2i 137	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦)
16	11, 15	nsyli 157	. . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
17	16	con2d 134	. . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}))
18	17	ralrimiv 3102	. . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
19		ralnex 3167	. . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
20	18, 19	sylib 217	. 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
21	6, 20	mt2 199	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by:  elirr  9356  nd1  10343  nd2  10344  nd3  10345  axunnd  10352  axregndlem1  10358  axregndlem2  10359  axregnd  10360  elpotr  33757  exnel  33778  distel  33779  ruvALT  40168  ralndv1  44597