Theorem elirr 9578

Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elirr	⊢ ¬ 𝐴 ∈ 𝐴

Proof of Theorem elirr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2	1, 1	eleq12d 2855	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	2	notbid 321	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
4		elirrv 9575	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
5	3, 4	vtoclg 3518	. 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴)
6		pm2.01 190	. 2 ⊢ ((𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴)
7	5, 6	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145

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-ext 2733  ax-sep 5249  ax-reg 9570

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836

This theorem is used by:  elneq  9579  nelaneq  9580  nelaneqOLD  9581  sucprcreg  9584  sucprcregOLD  9585  ruv  9586  disjcsn  9588  prv1n  36165  rankeq1o  36902  hfninf  36905  suceldisj  39718