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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 8326 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 [cec 8281 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ec 8285 |
This theorem is referenced by: ecid 8356 sylow2alem2 18737 sylow2a 18738 sylow2blem1 18739 efgval2 18844 efgrelexlemb 18870 efgcpbllemb 18875 frgpnabllem1 18987 tgpconncomp 22715 qustgphaus 22725 vitalilem2 24204 vitalilem3 24205 isbndx 35054 prtlem10 35995 prtlem19 36008 prter3 36012 |
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