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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 8788 | . 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 2106 Vcvv 3478 class class class wbr 5148 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: ecid 8821 sylow2alem2 19651 sylow2a 19652 sylow2blem1 19653 efgval2 19757 efgrelexlemb 19783 efgcpbllemb 19788 frgpnabllem1 19906 tgpconncomp 24137 qustgphaus 24147 vitalilem2 25658 vitalilem3 25659 isbndx 37769 prtlem10 38847 prtlem19 38860 prter3 38864 |
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