Theorem eirrv 3449

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 3452 and efrirr 2180, but this proof is direct from the Axiom of Regularity.)

Assertion

Ref	Expression
eirrv	⊢ ¬ x ∈ x

Proof of Theorem eirrv

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . 4 ⊢ (y = x → (y ∈ {x} ↔ x ∈ {x}))
2		visset 1350	. . . . 5 ⊢ x ∈ V
3	2	snid 1830	. . . 4 ⊢ x ∈ {x}
4	1, 3	a4w1 930	. . 3 ⊢ ∃y y ∈ {x}
5		snex 1859	. . . 4 ⊢ {x} ∈ V
6	5	zfregcl 3446	. . 3 ⊢ (∃y y ∈ {x} → ∃y ∈ {x}∀z ∈ y ¬ z ∈ {x})
7	4, 6	ax-mp 6	. 2 ⊢ ∃y ∈ {x}∀z ∈ y ¬ z ∈ {x}
8		ax-14 805	. . . . . . . . 9 ⊢ (x = y → (x ∈ x → x ∈ y))
9	8	eqcoms 813	. . . . . . . 8 ⊢ (y = x → (x ∈ x → x ∈ y))
10	9	com12 13	. . . . . . 7 ⊢ (x ∈ x → (y = x → x ∈ y))
11		elsn 1820	. . . . . . 7 ⊢ (y ∈ {x} ↔ y = x)
12	10, 11	syl5ib 181	. . . . . 6 ⊢ (x ∈ x → (y ∈ {x} → x ∈ y))
13		eleq1 1149	. . . . . . . . 9 ⊢ (z = x → (z ∈ {x} ↔ x ∈ {x}))
14	13	negbid 463	. . . . . . . 8 ⊢ (z = x → (¬ z ∈ {x} ↔ ¬ x ∈ {x}))
15	14	rcla4v 1402	. . . . . . 7 ⊢ (∀z ∈ y ¬ z ∈ {x} → (x ∈ y → ¬ x ∈ {x}))
16	3, 15	mt2i 97	. . . . . 6 ⊢ (∀z ∈ y ¬ z ∈ {x} → ¬ x ∈ y)
17	12, 16	nsyli 106	. . . . 5 ⊢ (x ∈ x → (∀z ∈ y ¬ z ∈ {x} → ¬ y ∈ {x}))
18	17	con2d 83	. . . 4 ⊢ (x ∈ x → (y ∈ {x} → ¬ ∀z ∈ y ¬ z ∈ {x}))
19	18	r19.21aiv 1259	. . 3 ⊢ (x ∈ x → ∀y ∈ {x} ¬ ∀z ∈ y ¬ z ∈ {x})
20		ralnex 1209	. . 3 ⊢ (∀y ∈ {x} ¬ ∀z ∈ y ¬ z ∈ {x} ↔ ¬ ∃y ∈ {x}∀z ∈ y ¬ z ∈ {x})
21	19, 20	sylib 173	. 2 ⊢ (x ∈ x → ¬ ∃y ∈ {x}∀z ∈ y ¬ z ∈ {x})
22	7, 21	mt2 96	1 ⊢ ¬ x ∈ x

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = weq 797   ∈ wel 803   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {csn 1808

This theorem is referenced by:  eirr 3450  aceq6b 3565  nd1 3732  nd2 3733  nd3 3734  axunnd 3742  axregndlem1 3748  axregndlem2 3749  axregnd 3750

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-reg 1078

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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