Theorem 0er 8679

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 5749	. 2 ⊢ Rel ∅
2		df-br 5080	. . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3		noel 4273	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
4	3	pm2.21i 119	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
5	2, 4	sylbi 218	. 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
6	3	pm2.21i 119	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
7	2, 6	sylbi 218	. . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
8	7	adantr 481	. 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧)
9		noel 4273	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
10		noel 4273	. . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
11	9, 10	2false 376	. . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12		df-br 5080	. . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
13	11, 12	bitr4i 279	. 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
14	1, 5, 8, 13	iseri 8668	1 ⊢ ∅ Er ∅

Syntax hints:   ∈ wcel 2119  ∅c0 4268  ⟨cop 4568   class class class wbr 5079   Er wer 8637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-er 8640

This theorem is referenced by: (None)