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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 7785 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 [cec 7740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 |
This theorem is referenced by: ecid 7812 sylow2alem2 18033 sylow2a 18034 sylow2blem1 18035 efgval2 18137 efgrelexlemb 18163 efgcpbllemb 18168 frgpnabllem1 18276 tgpconncomp 21916 qustgphaus 21926 vitalilem2 23378 vitalilem3 23379 isbndx 33581 prtlem10 34150 prtlem19 34163 prter3 34167 |
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