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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 3501 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V) | |
| 2 | elfix2.1 | . . 3 ⊢ Rel 𝑅 | |
| 3 | 2 | brrelex1i 5741 | . 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V) |
| 4 | eleq1 2829 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅)) | |
| 5 | breq12 5148 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
| 6 | 5 | anidms 566 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) |
| 7 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 8 | 7 | elfix 35904 | . . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥) |
| 9 | 4, 6, 8 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)) |
| 10 | 1, 3, 9 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 Rel wrel 5690 Fix cfix 35836 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-dm 5695 df-fix 35860 |
| This theorem is referenced by: (None) |
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