Theorem 0er 6814

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 4882	. . . 4 ⊢ Rel ∅
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ∅)
3		df-br 4115	. . . . 5 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4		noel 3516	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
5	4	pm2.21i 651	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦∅𝑥)
6	3, 5	sylbi 121	. . . 4 ⊢ (𝑥∅𝑦 → 𝑦∅𝑥)
7	6	adantl 277	. . 3 ⊢ ((⊤ ∧ 𝑥∅𝑦) → 𝑦∅𝑥)
8	4	pm2.21i 651	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥∅𝑧)
9	3, 8	sylbi 121	. . . 4 ⊢ (𝑥∅𝑦 → 𝑥∅𝑧)
10	9	ad2antrl 490	. . 3 ⊢ ((⊤ ∧ (𝑥∅𝑦 ∧ 𝑦∅𝑧)) → 𝑥∅𝑧)
11		noel 3516	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
12		noel 3516	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
13	11, 12	2false 709	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14		df-br 4115	. . . . 5 ⊢ (𝑥∅𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
15	13, 14	bitr4i 187	. . . 4 ⊢ (𝑥 ∈ ∅ ↔ 𝑥∅𝑥)
16	15	a1i 9	. . 3 ⊢ (⊤ → (𝑥 ∈ ∅ ↔ 𝑥∅𝑥))
17	2, 7, 10, 16	iserd 6806	. 2 ⊢ (⊤ → ∅ Er ∅)
18	17	mptru 1407	1 ⊢ ∅ Er ∅

Syntax hints:   ↔ wb 105  ⊤wtru 1399   ∈ wcel 2205  ∅c0 3512  ⟨cop 3697   class class class wbr 4114  Rel wrel 4759   Er wer 6777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-er 6780

This theorem is referenced by: (None)