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| 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 5809 | . 2 ⊢ Rel ∅ | |
| 2 | df-br 5144 | . . 3 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
| 3 | noel 4338 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
| 4 | 3 | pm2.21i 119 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) | 
| 5 | 2, 4 | sylbi 217 | . 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) | 
| 6 | 3 | pm2.21i 119 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) | 
| 7 | 2, 6 | sylbi 217 | . . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) | 
| 8 | 7 | adantr 480 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) | 
| 9 | noel 4338 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
| 10 | noel 4338 | . . . 4 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
| 11 | 9, 10 | 2false 375 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) | 
| 12 | df-br 5144 | . . 3 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
| 13 | 11, 12 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) | 
| 14 | 1, 5, 8, 13 | iseri 8772 | 1 ⊢ ∅ Er ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ∅c0 4333 〈cop 4632 class class class wbr 5143 Er wer 8742 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-er 8745 | 
| This theorem is referenced by: (None) | 
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