Theorem elirrvOLDOLD 9620

Description: Obsolete version of elirrv 9618 as of 27-Dec-2025. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elirrvOLDOLD	⊢ ¬ 𝑥 ∈ 𝑥

Proof of Theorem elirrvOLDOLD

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

Step	Hyp	Ref	Expression
1		vsnex 5397	. . 3 ⊢ {𝑥} ∈ V
2		eleq1w 2847	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3		vsnid 4624	. . . 4 ⊢ 𝑥 ∈ {𝑥}
4	2, 3	speivw 2008	. . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥}
5		zfregcl 9615	. . 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 4600	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8		ax9 2161	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
9	8	equcoms 2055	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
10	9	com12 33	. . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦))
11	7, 10	biimtrid 245	. . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦))
12		eleq1w 2847	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
13	12	notbid 321	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
14	13	rspccv 3577	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥}))
15	3, 14	mt2i 138	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦)
16	11, 15	nsyli 158	. . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
17	16	con2d 135	. . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}))
18	17	ralrimiv 3157	. . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
19		ralnex 3092	. . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
20	18, 19	sylib 221	. 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
21	6, 20	mt2 203	1 ⊢ ¬ 𝑥 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454  {csn 4588

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-reg 9613

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by: (None)