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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 5665 | . 2 ⊢ Rel ∅ | |
2 | df-br 5058 | . . 3 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
3 | noel 4293 | . . . 4 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
4 | 3 | pm2.21i 119 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
5 | 2, 4 | sylbi 218 | . 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
6 | 3 | pm2.21i 119 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
7 | 2, 6 | sylbi 218 | . . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
8 | 7 | adantr 481 | . 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧) |
9 | noel 4293 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4293 | . . . 4 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
11 | 9, 10 | 2false 377 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
12 | df-br 5058 | . . 3 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
13 | 11, 12 | bitr4i 279 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8305 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∅c0 4288 〈cop 4563 class class class wbr 5057 Er wer 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-er 8278 |
This theorem is referenced by: (None) |
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