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Mirrors > Home > MPE Home > Th. List > 0er | Structured version Visualization version GIF version |
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 1-May-2021.) |
Ref | Expression |
---|---|
0er | ⊢ ∅ Er ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 5801 | . 2 ⊢ Rel ∅ | |
2 | df-br 5150 | . . 3 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
3 | noel 4330 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
5 | 2, 4 | sylbi 216 | . 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
6 | 3 | pm2.21i 119 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
7 | 2, 6 | sylbi 216 | . . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
8 | 7 | adantr 479 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) |
9 | noel 4330 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4330 | . . . 4 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
11 | 9, 10 | 2false 374 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
12 | df-br 5150 | . . 3 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
13 | 11, 12 | bitr4i 277 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8752 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∅c0 4322 〈cop 4636 class class class wbr 5149 Er wer 8722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-er 8725 |
This theorem is referenced by: (None) |
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