Theorem elirrv 9558

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 9572 and efrirr 5641 (see elirrvALT 9573), but this proof is direct from ax-reg 9553. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9553 directly. (Revised by BTernaryTau, 27-Dec-2025.) Avoid ax-pr 5404. (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 2148	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑥 ↔ 𝑥 ∈ 𝑥))
2	1	biimprcd 253	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑥 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑥))
3	2	pm4.71rd 571	. . . . . . 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 1948	. . . . 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 1997	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝑥
8		exbi 1875	. . . . . . . 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 9553	. . . . . . 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 2148	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
14		elequ1 2148	. . . . . . . . . . . . . 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 2016	. . . . . . . . . . 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 629	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)) → (𝑧 ∈ 𝑦 → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)))
20	19	ex 417	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → (∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦) → (𝑧 ∈ 𝑦 → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))))
21	20	impcomd 416	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥) → ((𝑧 ∈ 𝑦 ∧ ∀𝑥(𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑦)) → (𝑧 = 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)))
22	21	aleximi 1860	. . . . . 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 2159	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 = 𝑥) → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑥))
25	24	anidms 576	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑥))
26	25	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑥))
27	26	equsexvw 2033	. . . . 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 5257	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 = 𝑥))
32	30, 31	exlimiiv 1959	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566  ∃wex 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-sep 5256  ax-reg 9553

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808

This theorem is referenced by:  elirr  9561  nd1  10571  nd2  10572  nd3  10573  axunnd  10580  axregndlem1  10586  axregndlem2  10587  axregnd  10588  axsepg2  35507  axsepg4  35510  axnulg  35512  axpowg2  35514  axpowg3  35515  elpotr  36225  exnel  36246  distel  36247  axtcond  36933  mh-setindnd  36992  ruvALT  43349  onsupmaxb  43914  ralndv1  47787