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Mirrors > Home > MPE Home > Th. List > 0erOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of 0er 7667 as of 1-May-2021. The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0erOLD | ⊢ ∅ Er ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rel0 5166 | . . . 4 ⊢ Rel ∅ | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Rel ∅) |
3 | df-br 4584 | . . . . 5 ⊢ (𝑥∅𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ∅) | |
4 | noel 3878 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑦〉 ∈ ∅ | |
5 | 4 | pm2.21i 115 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑦∅𝑥) |
6 | 3, 5 | sylbi 206 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥) |
7 | 6 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥) |
8 | 4 | pm2.21i 115 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∅ → 𝑥∅𝑧) |
9 | 3, 8 | sylbi 206 | . . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧) |
10 | 9 | ad2antrl 760 | . . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧) |
11 | noel 3878 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
12 | noel 3878 | . . . . . 6 ⊢ ¬ 〈𝑥, 𝑥〉 ∈ ∅ | |
13 | 11, 12 | 2false 364 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↔ 〈𝑥, 𝑥〉 ∈ ∅) |
14 | df-br 4584 | . . . . 5 ⊢ (𝑥∅𝑥 ↔ 〈𝑥, 𝑥〉 ∈ ∅) | |
15 | 13, 14 | bitr4i 266 | . . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥) |
16 | 15 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)) |
17 | 2, 7, 10, 16 | iserd 7655 | . 2 ⊢ (⊤ → ∅ Er ∅) |
18 | 17 | trud 1484 | 1 ⊢ ∅ Er ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ⊤wtru 1476 ∈ wcel 1977 ∅c0 3874 〈cop 4131 class class class wbr 4583 Rel wrel 5043 Er wer 7626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-er 7629 |
This theorem is referenced by: (None) |
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