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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 3514 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V) | |
2 | elfix2.1 | . . 3 ⊢ Rel 𝑅 | |
3 | 2 | brrelex1i 5610 | . 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V) |
4 | eleq1 2902 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅)) | |
5 | breq12 5073 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
6 | 5 | anidms 569 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴)) |
7 | vex 3499 | . . . 4 ⊢ 𝑥 ∈ V | |
8 | 7 | elfix 33366 | . . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥) |
9 | 4, 6, 8 | vtoclbg 3571 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)) |
10 | 1, 3, 9 | pm5.21nii 382 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 class class class wbr 5068 Rel wrel 5562 Fix cfix 33298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-dm 5567 df-fix 33322 |
This theorem is referenced by: (None) |
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