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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 5671 | . 2 ⊢ Rel ∅ | |
2 | df-br 5066 | . . 3 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
3 | noel 4295 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
5 | 2, 4 | sylbi 219 | . 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
6 | 3 | pm2.21i 119 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
7 | 2, 6 | sylbi 219 | . . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
8 | 7 | adantr 483 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) |
9 | noel 4295 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4295 | . . . 4 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
11 | 9, 10 | 2false 378 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
12 | df-br 5066 | . . 3 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
13 | 11, 12 | bitr4i 280 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8315 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∅c0 4290 〈cop 4572 class class class wbr 5065 Er wer 8285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-er 8288 |
This theorem is referenced by: (None) |
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