Theorem 0er 8782

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.)

Assertion

Ref	Expression
0er	⊢ ∅ Er ∅

Proof of Theorem 0er

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5812	. 2 ⊢ Rel ∅
2		df-br 5149	. . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3		noel 4344	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
4	3	pm2.21i 119	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
5	2, 4	sylbi 217	. 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
6	3	pm2.21i 119	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
7	2, 6	sylbi 217	. . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
8	7	adantr 480	. 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧)
9		noel 4344	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
10		noel 4344	. . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
11	9, 10	2false 375	. . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12		df-br 5149	. . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
13	11, 12	bitr4i 278	. 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
14	1, 5, 8, 13	iseri 8771	1 ⊢ ∅ Er ∅

Syntax hints:   ∈ wcel 2106  ∅c0 4339  ⟨cop 4637   class class class wbr 5148   Er wer 8741

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-er 8744

This theorem is referenced by: (None)