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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 9053. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
cnvepnep | ⊢ (◡ E ∩ E ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eprel 5430 | . . . . . 6 ⊢ E = {〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} | |
2 | 1 | cnveqi 5709 | . . . . 5 ⊢ ◡ E = ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} |
3 | cnvopab 5964 | . . . . 5 ⊢ ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} | |
4 | 2, 3 | eqtri 2821 | . . . 4 ⊢ ◡ E = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} |
5 | df-eprel 5430 | . . . 4 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
6 | 4, 5 | ineq12i 4137 | . . 3 ⊢ (◡ E ∩ E ) = ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
7 | inopab 5665 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} | |
8 | 6, 7 | eqtri 2821 | . 2 ⊢ (◡ E ∩ E ) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} |
9 | en2lp 9053 | . . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) | |
10 | 9 | gen2 1798 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) |
11 | opab0 5406 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) | |
12 | 10, 11 | mpbir 234 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ |
13 | 8, 12 | eqtri 2821 | 1 ⊢ (◡ E ∩ E ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∀wal 1536 = wceq 1538 ∩ cin 3880 ∅c0 4243 {copab 5092 E cep 5429 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-reg 9040 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-fr 5478 df-xp 5525 df-rel 5526 df-cnv 5527 |
This theorem is referenced by: epnsym 9056 |
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