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Mirrors > Home > MPE Home > Th. List > epelg | Structured version Visualization version GIF version |
Description: The epsilon relation and membership are the same. General version of epel 5032. (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 4654 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | elopab 4983 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦)) | |
3 | vex 3203 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
4 | vex 3203 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | pm3.2i 471 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
6 | opeqex 4962 | . . . . . . . . . 10 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
7 | 5, 6 | mpbiri 248 | . . . . . . . . 9 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | simpld 475 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
9 | 8 | adantr 481 | . . . . . . 7 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
10 | 9 | exlimivv 1860 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
11 | 2, 10 | sylbi 207 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V) |
12 | df-eprel 5029 | . . . . 5 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
13 | 11, 12 | eleq2s 2719 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
14 | 1, 13 | sylbi 207 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
16 | elex 3212 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
17 | 16 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
18 | eleq12 2691 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
19 | 18, 12 | brabga 4989 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
20 | 19 | expcom 451 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
21 | 15, 17, 20 | pm5.21ndd 369 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 class class class wbr 4653 {copab 4712 E cep 5028 |
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-ne 2795 df-rab 2921 df-v 3202 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-eprel 5029 |
This theorem is referenced by: epelc 5031 efrirr 5095 efrn2lp 5096 predep 5706 epne3 6980 cnfcomlem 8596 fpwwe2lem6 9457 ltpiord 9709 orvcelval 30530 brcnvep 34029 |
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