Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > epelg | GIF version |
Description: The epsilon relation and membership are the same. General version of epel 4047. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
epelg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3786 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | elopab 4013 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦)) | |
3 | vex 2604 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
4 | vex 2604 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | pm3.2i 266 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
6 | opeqex 4004 | . . . . . . . . . 10 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
7 | 5, 6 | mpbiri 166 | . . . . . . . . 9 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | simpld 110 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
9 | 8 | adantr 270 | . . . . . . 7 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
10 | 9 | exlimivv 1817 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
11 | 2, 10 | sylbi 119 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V) |
12 | df-eprel 4044 | . . . . 5 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
13 | 11, 12 | eleq2s 2173 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
14 | 1, 13 | sylbi 119 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
15 | 14 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
16 | elex 2610 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
17 | 16 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
18 | eleq12 2143 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
19 | 18, 12 | brabga 4019 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
20 | 19 | expcom 114 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
21 | 15, 17, 20 | pm5.21ndd 653 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 〈cop 3401 class class class wbr 3785 {copab 3838 E cep 4042 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-eprel 4044 |
This theorem is referenced by: epelc 4046 efrirr 4108 smoiso 5940 ecidg 6193 ordiso2 6446 ltpiord 6509 |
Copyright terms: Public domain | W3C validator |