Theorem 0er 8712

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 5769	. 2 ⊢ Rel ∅
2		df-br 5100	. . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3		noel 4290	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
4	3	pm2.21i 119	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
5	2, 4	sylbi 219	. 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
6	3	pm2.21i 119	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
7	2, 6	sylbi 219	. . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
8	7	adantr 484	. 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧)
9		noel 4290	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
10		noel 4290	. . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
11	9, 10	2false 377	. . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12		df-br 5100	. . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
13	11, 12	bitr4i 280	. 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
14	1, 5, 8, 13	iseri 8701	1 ⊢ ∅ Er ∅

Syntax hints:   ∈ wcel 2141  ∅c0 4285  ⟨cop 4587   class class class wbr 5099   Er wer 8670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-er 8673

This theorem is referenced by: (None)