Theorem elirrvOLD 9484

Description: Obsolete version of elirrv 9483 as of 27-Dec-2025. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elirrvOLD	⊢ ¬ 𝑥 ∈ 𝑥

Proof of Theorem elirrvOLD

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

Step	Hyp	Ref	Expression
1		vsnex 5370	. . 3 ⊢ {𝑥} ∈ V
2		eleq1w 2814	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
3		vsnid 4613	. . . 4 ⊢ 𝑥 ∈ {𝑥}
4	2, 3	speivw 1974	. . 3 ⊢ ∃𝑦 𝑦 ∈ {𝑥}
5		zfregcl 9480	. . 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 4589	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
8		ax9 2125	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
9	8	equcoms 2021	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 ∈ 𝑥 → 𝑥 ∈ 𝑦))
10	9	com12 32	. . . . . . 7 ⊢ (𝑥 ∈ 𝑥 → (𝑦 = 𝑥 → 𝑥 ∈ 𝑦))
11	7, 10	biimtrid 242	. . . . . 6 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → 𝑥 ∈ 𝑦))
12		eleq1w 2814	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑥} ↔ 𝑥 ∈ {𝑥}))
13	12	notbid 318	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (¬ 𝑧 ∈ {𝑥} ↔ ¬ 𝑥 ∈ {𝑥}))
14	13	rspccv 3569	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → (𝑥 ∈ 𝑦 → ¬ 𝑥 ∈ {𝑥}))
15	3, 14	mt2i 137	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑥 ∈ 𝑦)
16	11, 15	nsyli 157	. . . . 5 ⊢ (𝑥 ∈ 𝑥 → (∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} → ¬ 𝑦 ∈ {𝑥}))
17	16	con2d 134	. . . 4 ⊢ (𝑥 ∈ 𝑥 → (𝑦 ∈ {𝑥} → ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥}))
18	17	ralrimiv 3123	. . 3 ⊢ (𝑥 ∈ 𝑥 → ∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
19		ralnex 3058	. . 3 ⊢ (∀𝑦 ∈ {𝑥} ¬ ∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥} ↔ ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
20	18, 19	sylib 218	. 2 ⊢ (𝑥 ∈ 𝑥 → ¬ ∃𝑦 ∈ {𝑥}∀𝑧 ∈ 𝑦 ¬ 𝑧 ∈ {𝑥})
21	6, 20	mt2 200	1 ⊢ ¬ 𝑥 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-v 3438  df-un 3902  df-sn 4574  df-pr 4576

This theorem is referenced by: (None)