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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 9071. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
cnvepnep | ⊢ (◡ E ∩ E ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eprel 5467 | . . . . . 6 ⊢ E = {〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} | |
2 | 1 | cnveqi 5747 | . . . . 5 ⊢ ◡ E = ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} |
3 | cnvopab 5999 | . . . . 5 ⊢ ◡{〈𝑦, 𝑥〉 ∣ 𝑦 ∈ 𝑥} = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} | |
4 | 2, 3 | eqtri 2846 | . . . 4 ⊢ ◡ E = {〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} |
5 | df-eprel 5467 | . . . 4 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
6 | 4, 5 | ineq12i 4189 | . . 3 ⊢ (◡ E ∩ E ) = ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) |
7 | inopab 5703 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑦 ∈ 𝑥} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} | |
8 | 6, 7 | eqtri 2846 | . 2 ⊢ (◡ E ∩ E ) = {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} |
9 | en2lp 9071 | . . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) | |
10 | 9 | gen2 1797 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) |
11 | opab0 5443 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) | |
12 | 10, 11 | mpbir 233 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ |
13 | 8, 12 | eqtri 2846 | 1 ⊢ (◡ E ∩ E ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∀wal 1535 = wceq 1537 ∩ cin 3937 ∅c0 4293 {copab 5130 E cep 5466 ◡ccnv 5556 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-reg 9058 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-eprel 5467 df-fr 5516 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: epnsym 9074 |
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