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Mirrors > Home > MPE Home > Th. List > epinid0 | Structured version Visualization version GIF version |
Description: The membership (epsilon) relation and the identity relation are disjoint. Variable-free version of nelaneq 9062. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) |
Ref | Expression |
---|---|
epinid0 | ⊢ ( E ∩ I ) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eprel 5464 | . . 3 ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} | |
2 | df-id 5459 | . . 3 ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | ineq12i 4186 | . 2 ⊢ ( E ∩ I ) = ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) |
4 | inopab 5700 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} ∩ {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} | |
5 | nelaneq 9062 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦) | |
6 | 5 | gen2 1793 | . . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦) |
7 | opab0 5440 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)) | |
8 | 6, 7 | mpbir 233 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝑦 ∧ 𝑥 = 𝑦)} = ∅ |
9 | 3, 4, 8 | 3eqtri 2848 | 1 ⊢ ( E ∩ I ) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∀wal 1531 = wceq 1533 ∩ cin 3934 ∅c0 4290 {copab 5127 I cid 5458 E cep 5463 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-reg 9055 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-id 5459 df-eprel 5464 df-xp 5560 df-rel 5561 |
This theorem is referenced by: (None) |
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