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Theorem 0er 8736
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 5797 . 2 Rel ∅
2 df-br 5148 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3 noel 4329 . . . 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 481 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
9 noel 4329 . . . 4 ¬ 𝑥 ∈ ∅
10 noel 4329 . . . 4 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
119, 102false 375 . . 3 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12 df-br 5148 . . 3 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1311, 12bitr4i 277 . 2 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
141, 5, 8, 13iseri 8726 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  c0 4321  cop 4633   class class class wbr 5147   Er wer 8696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-er 8699
This theorem is referenced by: (None)
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