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Mirrors > Home > ILE Home > Th. List > 0er | GIF version |
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) |
Ref | Expression |
---|---|
0er | ⊢ ∅ Er ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 4751 | . . . 4 ⊢ Rel ∅ | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → Rel ∅) |
3 | df-br 4004 | . . . . 5 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
4 | noel 3426 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
5 | 4 | pm2.21i 646 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
6 | 3, 5 | sylbi 121 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
7 | 6 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥) |
8 | 4 | pm2.21i 646 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
9 | 3, 8 | sylbi 121 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
10 | 9 | ad2antrl 490 | . . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧) |
11 | noel 3426 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
12 | noel 3426 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
13 | 11, 12 | 2false 701 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
14 | df-br 4004 | . . . . 5 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
15 | 13, 14 | bitr4i 187 | . . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
16 | 15 | a1i 9 | . . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)) |
17 | 2, 7, 10, 16 | iserd 6560 | . 2 ⊢ (⊤ → ∅ Er ∅) |
18 | 17 | mptru 1362 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ⊤wtru 1354 ∈ wcel 2148 ∅c0 3422 〈cop 3595 class class class wbr 4003 Rel wrel 4631 Er wer 6531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-er 6534 |
This theorem is referenced by: (None) |
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