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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfix2 | Structured version Visualization version GIF version |
Description: Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
elfix2.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
elfix2 | ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3416 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V) | |
2 | elfix2.1 | . . 3 ⊢ Rel 𝑅 | |
3 | 2 | brrelex1i 5590 | . 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V) |
4 | eleq1 2818 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅)) | |
5 | breq12 5044 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
6 | 5 | anidms 570 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) |
7 | vex 3402 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | 7 | elfix 33891 | . . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥) |
9 | 4, 6, 8 | vtoclbg 3473 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)) |
10 | 1, 3, 9 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2112 Vcvv 3398 class class class wbr 5039 Rel wrel 5541 Fix cfix 33823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-dm 5546 df-fix 33847 |
This theorem is referenced by: (None) |
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