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Mirrors > Home > ILE Home > Th. List > 0er | GIF version |
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
0er | ⊢ ∅ Er ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4723 | . . . 4 ⊢ Rel ∅ | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → Rel ∅) |
3 | df-br 3977 | . . . . 5 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
4 | noel 3408 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
5 | 4 | pm2.21i 636 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
6 | 3, 5 | sylbi 120 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
7 | 6 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥) |
8 | 4 | pm2.21i 636 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
9 | 3, 8 | sylbi 120 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
10 | 9 | ad2antrl 482 | . . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧) |
11 | noel 3408 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
12 | noel 3408 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
13 | 11, 12 | 2false 691 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
14 | df-br 3977 | . . . . 5 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
15 | 13, 14 | bitr4i 186 | . . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
16 | 15 | a1i 9 | . . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)) |
17 | 2, 7, 10, 16 | iserd 6518 | . 2 ⊢ (⊤ → ∅ Er ∅) |
18 | 17 | mptru 1351 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ⊤wtru 1343 ∈ wcel 2135 ∅c0 3404 〈cop 3573 class class class wbr 3976 Rel wrel 4603 Er wer 6489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-er 6492 |
This theorem is referenced by: (None) |
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