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Theorem 0er 8762
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 5801 . 2 Rel ∅
2 df-br 5150 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3 noel 4330 . . . 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 479 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
9 noel 4330 . . . 4 ¬ 𝑥 ∈ ∅
10 noel 4330 . . . 4 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
119, 102false 374 . . 3 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12 df-br 5150 . . 3 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1311, 12bitr4i 277 . 2 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
141, 5, 8, 13iseri 8752 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  c0 4322  cop 4636   class class class wbr 5149   Er wer 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-er 8725
This theorem is referenced by: (None)
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