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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 3414 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V) | |
2 | elfix2.1 | . . 3 ⊢ Rel 𝑅 | |
3 | 2 | brrelex1i 5406 | . 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V) |
4 | eleq1 2847 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅)) | |
5 | breq12 4891 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
6 | 5 | anidms 562 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) |
7 | vex 3401 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | 7 | elfix 32599 | . . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥) |
9 | 4, 6, 8 | vtoclbg 3468 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)) |
10 | 1, 3, 9 | pm5.21nii 370 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 Rel wrel 5360 Fix cfix 32531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-dm 5365 df-fix 32555 |
This theorem is referenced by: (None) |
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