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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 5756 | . 2 ⊢ Rel ∅ | |
2 | df-br 5107 | . . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) | |
3 | noel 4291 | . . . 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 482 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) |
9 | noel 4291 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4291 | . . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅ | |
11 | 9, 10 | 2false 376 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) |
12 | df-br 5107 | . . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅) | |
13 | 11, 12 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8676 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∅c0 4283 ⟨cop 4593 class class class wbr 5106 Er wer 8646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-er 8649 |
This theorem is referenced by: (None) |
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