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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 9646. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
| Ref | Expression |
|---|---|
| cnvepnep | ⊢ (◡ E ∩ E ) = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-eprel 5584 | . . . . . 6 ⊢ E = {〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} | |
| 2 | 1 | cnveqi 5885 | . . . . 5 ⊢ ◡ E = ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} |
| 3 | cnvopab 6157 | . . . . 5 ⊢ ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} | |
| 4 | 2, 3 | eqtri 2765 | . . . 4 ⊢ ◡ E = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} |
| 5 | df-eprel 5584 | . . . 4 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
| 6 | 4, 5 | ineq12i 4218 | . . 3 ⊢ (◡ E ∩ E ) = ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
| 7 | inopab 5839 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} | |
| 8 | 6, 7 | eqtri 2765 | . 2 ⊢ (◡ E ∩ E ) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} |
| 9 | en2lp 9646 | . . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) | |
| 10 | 9 | gen2 1796 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) |
| 11 | opab0 5559 | . . 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 2765 | 1 ⊢ (◡ E ∩ E ) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 = wceq 1540 ∩ cin 3950 ∅c0 4333 {copab 5205 E cep 5583 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-fr 5637 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: epnsym 9649 |
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