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Mirrors > Home > ILE Home > Th. List > epelg | GIF version |
Description: The epsilon relation and membership are the same. General version of epel 4174. (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 3896 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | elopab 4140 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦)) | |
3 | vex 2660 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
4 | vex 2660 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | pm3.2i 268 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
6 | opeqex 4131 | . . . . . . . . . 10 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
7 | 5, 6 | mpbiri 167 | . . . . . . . . 9 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | simpld 111 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
9 | 8 | adantr 272 | . . . . . . 7 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
10 | 9 | exlimivv 1850 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
11 | 2, 10 | sylbi 120 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V) |
12 | df-eprel 4171 | . . . . 5 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
13 | 11, 12 | eleq2s 2209 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
14 | 1, 13 | sylbi 120 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
15 | 14 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
16 | elex 2668 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
17 | 16 | a1i 9 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
18 | eleq12 2179 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
19 | 18, 12 | brabga 4146 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
20 | 19 | expcom 115 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
21 | 15, 17, 20 | pm5.21ndd 677 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 〈cop 3496 class class class wbr 3895 {copab 3948 E cep 4169 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-eprel 4171 |
This theorem is referenced by: epelc 4173 efrirr 4235 smoiso 6153 ecidg 6447 ordiso2 6872 ltpiord 7075 |
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