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| Mirrors > Home > MPE Home > Th. List > elec | Structured version Visualization version GIF version | ||
| Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) |
| Ref | Expression |
|---|---|
| elec.1 | ⊢ 𝐴 ∈ V |
| elec.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| elec | ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elec.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | elec.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | elecg 8789 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 [cec 8743 |
| 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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 |
| This theorem is referenced by: ecid 8822 sylow2alem2 19636 sylow2a 19637 sylow2blem1 19638 efgval2 19742 efgrelexlemb 19768 efgcpbllemb 19773 frgpnabllem1 19891 tgpconncomp 24121 qustgphaus 24131 vitalilem2 25644 vitalilem3 25645 isbndx 37789 prtlem10 38866 prtlem19 38879 prter3 38883 |
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