![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > epelc | GIF version |
Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
epelc.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
epelc | ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epelc.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | epelg 4315 | . 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 Vcvv 2756 class class class wbr 4025 E cep 4312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2758 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-br 4026 df-opab 4087 df-eprel 4314 |
This theorem is referenced by: epel 4317 epini 5024 ecid 6639 ordiso2 7080 |
Copyright terms: Public domain | W3C validator |