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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 5565 | . 2 ⊢ Rel ∅ | |
2 | df-br 4969 | . . 3 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
3 | noel 4222 | . . . 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 4222 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
10 | noel 4222 | . . . 4 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
11 | 9, 10 | 2false 377 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
12 | df-br 4969 | . . 3 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
13 | 11, 12 | bitr4i 279 | . 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
14 | 1, 5, 8, 13 | iseri 8173 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2083 ∅c0 4217 〈cop 4484 class class class wbr 4968 Er wer 8143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-br 4969 df-opab 5031 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-er 8146 |
This theorem is referenced by: (None) |
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