Theorem 0er 6163

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 4480	. . . 4 ⊢ Rel ∅
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ∅)
3		df-br 3786	. . . . 5 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4		noel 3255	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
5	4	pm2.21i 607	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
6	3, 5	sylbi 119	. . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
7	6	adantl 271	. . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥)
8	4	pm2.21i 607	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
9	3, 8	sylbi 119	. . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
10	9	ad2antrl 473	. . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧)
11		noel 3255	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
12		noel 3255	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
13	11, 12	2false 649	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14		df-br 3786	. . . . 5 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
15	13, 14	bitr4i 185	. . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
16	15	a1i 9	. . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥))
17	2, 7, 10, 16	iserd 6155	. 2 ⊢ (⊤ → ∅ Er ∅)
18	17	trud 1293	1 ⊢ ∅ Er ∅

Syntax hints:   ↔ wb 103  ⊤wtru 1285   ∈ wcel 1433  ∅c0 3251  ⟨cop 3401   class class class wbr 3785  Rel wrel 4368   Er wer 6126

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-er 6129

This theorem is referenced by: (None)