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Theorem 0er 8686
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.
StepHypRef Expression
1 rel0 5756 . 2 Rel ∅
2 df-br 5107 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3 noel 4291 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
43pm2.21i 119 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
52, 4sylbi 216 . 2 (𝑥𝑦𝑦𝑥)
63pm2.21i 119 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
72, 6sylbi 216 . . 3 (𝑥𝑦𝑥𝑧)
87adantr 482 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
9 noel 4291 . . . 4 ¬ 𝑥 ∈ ∅
10 noel 4291 . . . 4 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
119, 102false 376 . . 3 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12 df-br 5107 . . 3 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1311, 12bitr4i 278 . 2 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
141, 5, 8, 13iseri 8676 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  c0 4283  cop 4593   class class class wbr 5106   Er wer 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-er 8649
This theorem is referenced by: (None)
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