Theorem elirr 9516

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 22	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2	1, 1	eleq12d 2831	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
3	2	notbid 318	. . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝐴 ∈ 𝐴))
4		elirrv 9514	. . 3 ⊢ ¬ 𝑥 ∈ 𝑥
5	3, 4	vtoclg 3513	. 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴)
6		pm2.01 188	. 2 ⊢ ((𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ 𝐴) → ¬ 𝐴 ∈ 𝐴)
7	5, 6	ax-mp 5	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812

This theorem is referenced by:  elneq  9517  nelaneq  9518  sucprcreg  9521  ruv  9522  disjcsn  9524  alephval3  10032  prv1n  35644  rankeq1o  36384  hfninf  36399  bj-iomnnom  37508  suceldisj  39063  omabs2  43683  setc1onsubc  49955