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Theorem 0er 8315
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 5665 . 2 Rel ∅
2 df-br 5058 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3 noel 4293 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
43pm2.21i 119 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
52, 4sylbi 218 . 2 (𝑥𝑦𝑦𝑥)
63pm2.21i 119 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
72, 6sylbi 218 . . 3 (𝑥𝑦𝑥𝑧)
87adantr 481 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
9 noel 4293 . . . 4 ¬ 𝑥 ∈ ∅
10 noel 4293 . . . 4 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
119, 102false 377 . . 3 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12 df-br 5058 . . 3 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1311, 12bitr4i 279 . 2 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
141, 5, 8, 13iseri 8305 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  c0 4288  cop 4563   class class class wbr 5057   Er wer 8275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-er 8278
This theorem is referenced by: (None)
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