Theorem elirrv 9572

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 9586 and efrirr 5639 (see elirrvALT 9587), but this proof is direct from ax-reg 9567. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9567 directly. (Revised by BTernaryTau, 27-Dec-2025.) Avoid ax-pr 5402. (Revised by BTernaryTau, 21-May-2026.) (Proof shortened by Matthew House, 23-May-2026.)

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		elequ1 2152	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑥 ↔ 𝑥 ∈ 𝑥))
2	1	biimprcd 253	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑥 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑥))
3	2	pm4.71rd 572	. . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑧 = 𝑥 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥)))
4	3	bibi2d 345	. . . . . 6 ⊢ (𝑥 ∈ 𝑥 → ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) ↔ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥))))
5	4	albidv 1953	. . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥))))
6	5	biimprcd 253	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥)) → (𝑥 ∈ 𝑥 → ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)))
7		ax6ev 2002	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝑥
8		exbi 1880	. . . . . . . 8 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → (∃𝑧 𝑧 ∈ 𝑦 ↔ ∃𝑧 𝑧 = 𝑥))
9	7, 8	mpbiri 261	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ∃𝑧 𝑧 ∈ 𝑦)
10		ax-reg 9567	. . . . . . 7 ⊢ (∃𝑧 𝑧 ∈ 𝑦 → ∃𝑧(𝑧 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)))
11	9, 10	syl 18	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ∃𝑧(𝑧 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)))
12		biimp 218	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → (𝑧 ∈ 𝑦 → 𝑧 = 𝑥))
13		elequ1 2152	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
14		elequ1 2152	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
15	14	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑦))
16	13, 15	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦) ↔ (𝑧 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑦)))
17	16	spvv 2021	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦) → (𝑧 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑦))
18	17	con2d 135	. . . . . . . . . 10 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦) → (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑧))
19	12, 18	anim12ii 630	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)) → (𝑧 ∈ 𝑦 → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)))
20	19	ex 418	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → (∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦) → (𝑧 ∈ 𝑦 → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))))
21	20	impcomd 417	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ((𝑧 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)) → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)))
22	21	aleximi 1865	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → (∃𝑧(𝑧 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)) → ∃𝑧(𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)))
23	11, 22	mpd 16	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ∃𝑧(𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))
24		elequ12 2163	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 = 𝑥) → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑥))
25	24	anidms 577	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑥))
26	25	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑥))
27	26	equsexvw 2038	. . . . 5 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧) ↔ ¬ 𝑥 ∈ 𝑥)
28	23, 27	sylib 221	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ¬ 𝑥 ∈ 𝑥)
29	6, 28	syl6 36	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥)) → (𝑥 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑥))
30	29	pm2.01d 192	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥)) → ¬ 𝑥 ∈ 𝑥)
31		axsepg 5256	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥))
32	30, 31	exlimiiv 1964	1 ⊢ ¬ 𝑥 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-sep 5255  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  elirr  9575  nd1  10599  nd2  10600  nd3  10601  axunnd  10608  axregndlem1  10614  axregndlem2  10615  axregnd  10616  axsepg2  35653  axsepg4  35656  axnulg  35658  axpowg2  35660  axpowg3  35661  elpotr  36345  exnel  36366  distel  36367  axtcond  37084  mh-setindnd  37143  ruvALT  43502  onsupmaxb  44067  ralndv1  47980