Theorem elfix2 36793

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 3474	. 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V)
2		elfix2.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5710	. 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V)
4		eleq1 2851	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅))
5		breq12 5110	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
6	5	anidms 577	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
7		vex 3457	. . . 4 ⊢ 𝑥 ∈ V
8	7	elfix 36792	. . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥)
9	4, 6, 8	vtoclbg 3522	. 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 1569   ∈ wcel 2146  Vcvv 3453   class class class wbr 5105  Rel wrel 5659   Fix cfix 36724

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-dm 5664  df-fix 36748

This theorem is used by: (None)