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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 9550. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.) |
Ref | Expression |
---|---|
cnvepnep | ⊢ (◡ E ∩ E ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eprel 5541 | . . . . . 6 ⊢ E = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} | |
2 | 1 | cnveqi 5834 | . . . . 5 ⊢ ◡ E = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} |
3 | cnvopab 6095 | . . . . 5 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} | |
4 | 2, 3 | eqtri 2761 | . . . 4 ⊢ ◡ E = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} |
5 | df-eprel 5541 | . . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} | |
6 | 4, 5 | ineq12i 4174 | . . 3 ⊢ (◡ E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}) |
7 | inopab 5789 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} | |
8 | 6, 7 | eqtri 2761 | . 2 ⊢ (◡ E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} |
9 | en2lp 9550 | . . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) | |
10 | 9 | gen2 1799 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦) |
11 | opab0 5515 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)) | |
12 | 10, 11 | mpbir 230 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ |
13 | 8, 12 | eqtri 2761 | 1 ⊢ (◡ E ∩ E ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∀wal 1540 = wceq 1542 ∩ cin 3913 ∅c0 4286 {copab 5171 E cep 5540 ◡ccnv 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-reg 9536 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-eprel 5541 df-fr 5592 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: epnsym 9553 |
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