![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cnvepnep | Structured version Visualization version GIF version |
Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9644. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
cnvepnep | ⊢ (◡ E ∩ E ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eprel 5589 | . . . . . 6 ⊢ E = {〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} | |
2 | 1 | cnveqi 5888 | . . . . 5 ⊢ ◡ E = ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} |
3 | cnvopab 6160 | . . . . 5 ⊢ ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} | |
4 | 2, 3 | eqtri 2763 | . . . 4 ⊢ ◡ E = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} |
5 | df-eprel 5589 | . . . 4 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
6 | 4, 5 | ineq12i 4226 | . . 3 ⊢ (◡ E ∩ E ) = ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
7 | inopab 5842 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} | |
8 | 6, 7 | eqtri 2763 | . 2 ⊢ (◡ E ∩ E ) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} |
9 | en2lp 9644 | . . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) | |
10 | 9 | gen2 1793 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) |
11 | opab0 5564 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) | |
12 | 10, 11 | mpbir 231 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ |
13 | 8, 12 | eqtri 2763 | 1 ⊢ (◡ E ∩ E ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1535 = wceq 1537 ∩ cin 3962 ∅c0 4339 {copab 5210 E cep 5588 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-eprel 5589 df-fr 5641 df-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: epnsym 9647 |
Copyright terms: Public domain | W3C validator |