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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 5795 | . 2 ⊢ Rel ∅ | |
2 | df-br 5143 | . . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) | |
3 | noel 4326 | . . . 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 480 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) |
9 | noel 4326 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4326 | . . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅ | |
11 | 9, 10 | 2false 375 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) |
12 | df-br 5143 | . . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) | |
13 | 11, 12 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8745 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ∅c0 4318 ⟨cop 4630 class class class wbr 5142 Er wer 8715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-er 8718 |
This theorem is referenced by: (None) |
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