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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 8691 | . 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 205 ∈ wcel 2106 Vcvv 3445 class class class wbr 5105 [cec 8646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-xp 5639 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ec 8650 |
This theorem is referenced by: ecid 8721 sylow2alem2 19400 sylow2a 19401 sylow2blem1 19402 efgval2 19506 efgrelexlemb 19532 efgcpbllemb 19537 frgpnabllem1 19651 tgpconncomp 23464 qustgphaus 23474 vitalilem2 24973 vitalilem3 24974 isbndx 36241 prtlem10 37327 prtlem19 37340 prter3 37344 |
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