![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5789 | . 2 ⊢ Rel ∅ | |
2 | df-br 5139 | . . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) | |
3 | noel 4322 | . . . 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 4322 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4322 | . . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅ | |
11 | 9, 10 | 2false 375 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) |
12 | df-br 5139 | . . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) | |
13 | 11, 12 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8726 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∅c0 4314 ⟨cop 4626 class class class wbr 5138 Er wer 8696 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-er 8699 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |