Theorem elirrv 9514

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 9525 and efrirr 5612 (see elirrvALT 9526), but this proof is direct from ax-reg 9509. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9509 directly. (Revised by BTernaryTau, 27-Dec-2025.)

Assertion

Ref	Expression
elirrv	⊢ ¬ 𝑥 ∈ 𝑥

Proof of Theorem elirrv

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

Step	Hyp	Ref	Expression
1		biimpr 220	. . . . 5 ⊢ ((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → (𝑦 = 𝑥 → 𝑦 ∈ 𝑤))
2	1	alimi 1813	. . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → ∀𝑦(𝑦 = 𝑥 → 𝑦 ∈ 𝑤))
3		elequ1 2121	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑤 ↔ 𝑥 ∈ 𝑤))
4	3	equsalvw 2006	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝑦 ∈ 𝑤) ↔ 𝑥 ∈ 𝑤)
5	2, 4	sylib 218	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → 𝑥 ∈ 𝑤)
6	3	equsexvw 2007	. . . . . . 7 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦 ∈ 𝑤) ↔ 𝑥 ∈ 𝑤)
7		exsimpr 1871	. . . . . . 7 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦 ∈ 𝑤) → ∃𝑦 𝑦 ∈ 𝑤)
8	6, 7	sylbir 235	. . . . . 6 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦 𝑦 ∈ 𝑤)
9		ax-reg 9509	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)))
10	8, 9	syl 17	. . . . 5 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)))
11		elequ1 2121	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
12		elequ1 2121	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑥 ∈ 𝑤))
13	12	notbid 318	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑤 ↔ ¬ 𝑥 ∈ 𝑤))
14	11, 13	imbi12d 344	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) ↔ (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑤)))
15	14	spvv 1990	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤) → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑤))
16		con2 135	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑤) → (𝑥 ∈ 𝑤 → ¬ 𝑥 ∈ 𝑦))
17	16	com12 32	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑤 → ((𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑤) → ¬ 𝑥 ∈ 𝑦))
18	17	anim2d 613	. . . . . . 7 ⊢ (𝑥 ∈ 𝑤 → ((𝑦 ∈ 𝑤 ∧ (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ 𝑤)) → (𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)))
19	15, 18	sylan2i 607	. . . . . 6 ⊢ (𝑥 ∈ 𝑤 → ((𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)) → (𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)))
20	19	eximdv 1919	. . . . 5 ⊢ (𝑥 ∈ 𝑤 → (∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤)) → ∃𝑦(𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)))
21	10, 20	mpd 15	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦))
22		19.29 1875	. . . . 5 ⊢ ((∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ ∃𝑦(𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)) → ∃𝑦((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)))
23		biimp 215	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → (𝑦 ∈ 𝑤 → 𝑦 = 𝑥))
24	23	anim1d 612	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → ((𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦) → (𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝑦)))
25		ax9v2 2127	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
26	25	equcoms 2022	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
27	26	con3dimp 408	. . . . . . . 8 ⊢ ((𝑦 = 𝑥 ∧ ¬ 𝑥 ∈ 𝑦) → ¬ 𝑥 ∈ 𝑥)
28	24, 27	syl6 35	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → ((𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦) → ¬ 𝑥 ∈ 𝑥))
29	28	imp 406	. . . . . 6 ⊢ (((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)) → ¬ 𝑥 ∈ 𝑥)
30	29	exlimiv 1932	. . . . 5 ⊢ (∃𝑦((𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ (𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)) → ¬ 𝑥 ∈ 𝑥)
31	22, 30	syl 17	. . . 4 ⊢ ((∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ ∃𝑦(𝑦 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑦)) → ¬ 𝑥 ∈ 𝑥)
32	21, 31	sylan2 594	. . 3 ⊢ ((∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) ∧ 𝑥 ∈ 𝑤) → ¬ 𝑥 ∈ 𝑥)
33	5, 32	mpdan 688	. 2 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥) → ¬ 𝑥 ∈ 𝑥)
34		el 5394	. . . 4 ⊢ ∃𝑤 𝑥 ∈ 𝑤
35	4	biimpri 228	. . . 4 ⊢ (𝑥 ∈ 𝑤 → ∀𝑦(𝑦 = 𝑥 → 𝑦 ∈ 𝑤))
36	34, 35	eximii 1839	. . 3 ⊢ ∃𝑤∀𝑦(𝑦 = 𝑥 → 𝑦 ∈ 𝑤)
37	36	sepexi 5248	. 2 ⊢ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ 𝑦 = 𝑥)
38	33, 37	exlimiiv 1933	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782

This theorem is referenced by:  elirr  9516  nd1  10510  nd2  10511  nd3  10512  axunnd  10519  axregndlem1  10525  axregndlem2  10526  axregnd  10527  axnulg  35283  elpotr  35992  exnel  36013  distel  36014  mh-setindnd  36686  ruvALT  43021  onsupmaxb  43590  ralndv1  47459