Theorem elirr 9597

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 2854	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	2	notbid 321	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
4		elirrv 9594	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
5	3, 4	vtoclg 3517	. 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 1569   ∈ wcel 2144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-reg 9589

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835

This theorem is used by:  elneq  9598  nelaneq  9599  nelaneqOLD  9600  sucprcreg  9603  sucprcregOLD  9604  ruv  9605  disjcsn  9607  prv1n  36282  rankeq1o  37024  hfninf  37027  suceldisj  39843