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Mirrors > Home > MPE Home > Th. List > epelgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of epelg 5466 as of 14-Jul-2023. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
epelgOLD | ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5067 | . . . 4 ⊢ (𝐴 E 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ E ) | |
2 | elopab 5414 | . . . . . 6 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦)) | |
3 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑥 ∈ V | |
4 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | pm3.2i 473 | . . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
6 | opeqex 5388 | . . . . . . . . . 10 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V))) | |
7 | 5, 6 | mpbiri 260 | . . . . . . . . 9 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
8 | 7 | simpld 497 | . . . . . . . 8 ⊢ (〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 → 𝐴 ∈ V) |
9 | 8 | adantr 483 | . . . . . . 7 ⊢ ((〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
10 | 9 | exlimivv 1933 | . . . . . 6 ⊢ (∃𝑥∃𝑦(〈𝐴, 𝐵〉 = 〈𝑥, 𝑦〉 ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V) |
11 | 2, 10 | sylbi 219 | . . . . 5 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V) |
12 | df-eprel 5465 | . . . . 5 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
13 | 11, 12 | eleq2s 2931 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ E → 𝐴 ∈ V) |
14 | 1, 13 | sylbi 219 | . . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V)) |
16 | elex 3512 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
17 | 16 | a1i 11 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V)) |
18 | eleq12 2902 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵)) | |
19 | 18, 12 | brabga 5421 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
20 | 19 | expcom 416 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))) |
21 | 15, 17, 20 | pm5.21ndd 383 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 {copab 5128 E cep 5464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-eprel 5465 |
This theorem is referenced by: (None) |
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