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Theorem 0er 7725
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 5204 . 2 Rel ∅
2 df-br 4614 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
3 noel 3895 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
43pm2.21i 116 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
52, 4sylbi 207 . 2 (𝑥𝑦𝑦𝑥)
63pm2.21i 116 . . . 4 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
72, 6sylbi 207 . . 3 (𝑥𝑦𝑥𝑧)
87adantr 481 . 2 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)
9 noel 3895 . . . 4 ¬ 𝑥 ∈ ∅
10 noel 3895 . . . 4 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
119, 102false 365 . . 3 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
12 df-br 4614 . . 3 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1311, 12bitr4i 267 . 2 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
141, 5, 8, 13iseri 7714 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  c0 3891  cop 4154   class class class wbr 4613   Er wer 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-er 7687
This theorem is referenced by: (None)
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