Theorem 0er 6626

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 4788	. . . 4 ⊢ Rel ∅
2	1	a1i 9	. . 3 ⊢ (⊤ → Rel ∅)
3		df-br 4034	. . . . 5 ⊢ (𝑥∅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4		noel 3454	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
5	4	pm2.21i 647	. . . . 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 647	. . . . 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 3454	. . . . . 6 ⊢ ¬ 𝑥 ∈ ∅
12		noel 3454	. . . . . 6 ⊢ ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
13	11, 12	2false 702	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14		df-br 4034	. . . . 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 6618	. 2 ⊢ (⊤ → ∅ Er ∅)
18	17	mptru 1373	1 ⊢ ∅ Er ∅

Syntax hints:   ↔ wb 105  ⊤wtru 1365   ∈ wcel 2167  ∅c0 3450  ⟨cop 3625   class class class wbr 4033  Rel wrel 4668   Er wer 6589

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-er 6592

This theorem is referenced by: (None)