Theorem elfix2 36881

Description: Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
elfix2.1	⊢ Rel 𝑅

Assertion

Ref	Expression
elfix2	⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Proof of Theorem elfix2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V)
2		elfix2.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5714	. 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V)
4		eleq1 2852	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅))
5		breq12 5113	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
6	5	anidms 577	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
7		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
8	7	elfix 36880	. . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥)
9	4, 6, 8	vtoclbg 3523	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴))
10	1, 3, 9	pm5.21nii 381	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  Rel wrel 5663   Fix cfix 36812

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 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-dm 5668  df-fix 36836

This theorem is used by: (None)