Theorem 0er 6393

Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

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 4602	. . . 4 ⊢ Rel ∅
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ∅)
3		df-br 3876	. . . . 5 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4		noel 3314	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
5	4	pm2.21i 615	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
6	3, 5	sylbi 120	. . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
7	6	adantl 273	. . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥)
8	4	pm2.21i 615	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
9	3, 8	sylbi 120	. . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
10	9	ad2antrl 477	. . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧)
11		noel 3314	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
12		noel 3314	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
13	11, 12	2false 658	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14		df-br 3876	. . . . 5 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
15	13, 14	bitr4i 186	. . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
16	15	a1i 9	. . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥))
17	2, 7, 10, 16	iserd 6385	. 2 ⊢ (⊤ → ∅ Er ∅)
18	17	mptru 1308	1 ⊢ ∅ Er ∅

Syntax hints:   ↔ wb 104  ⊤wtru 1300   ∈ wcel 1448  ∅c0 3310  ⟨cop 3477   class class class wbr 3875  Rel wrel 4482   Er wer 6356

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-er 6359

This theorem is referenced by: (None)