Theorem 0er 7780

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 5243	. 2 ⊢ Rel ∅
2		df-br 4654	. . 3 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3		noel 3919	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
4	3	pm2.21i 116	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
5	2, 4	sylbi 207	. 2 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
6	3	pm2.21i 116	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
7	2, 6	sylbi 207	. . 3 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
8	7	adantr 481	. 2 ⊢ ((𝑥∅𝑦 ∧ 𝑦∅𝑧) → 𝑥∅𝑧)
9		noel 3919	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
10		noel 3919	. . . 4 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
11	9, 10	2false 365	. . 3 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12		df-br 4654	. . 3 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
13	11, 12	bitr4i 267	. 2 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
14	1, 5, 8, 13	iseri 7769	1 ⊢ ∅ Er ∅

Syntax hints:   ∈ wcel 1990  ∅c0 3915  ⟨cop 4183   class class class wbr 4653   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by: (None)