Theorem 0er 8783

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 5809	. 2 ⊢ Rel ∅
2		df-br 5144	. . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3		noel 4338	. . . 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 4338	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
10		noel 4338	. . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
11	9, 10	2false 375	. . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12		df-br 5144	. . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
13	11, 12	bitr4i 278	. 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
14	1, 5, 8, 13	iseri 8772	1 ⊢ ∅ Er ∅

Syntax hints:   ∈ wcel 2108  ∅c0 4333  ⟨cop 4632   class class class wbr 5143   Er wer 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-er 8745

This theorem is referenced by: (None)