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Theorem 0erOLD 7668
 Description: Obsolete proof of 0er 7667 as of 1-May-2021. The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
0erOLD ∅ Er ∅

Proof of Theorem 0erOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rel0 5166 . . . 4 Rel ∅
21a1i 11 . . 3 (⊤ → Rel ∅)
3 df-br 4584 . . . . 5 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4 noel 3878 . . . . . 6 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
54pm2.21i 115 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
63, 5sylbi 206 . . . 4 (𝑥𝑦𝑦𝑥)
76adantl 481 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
84pm2.21i 115 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
93, 8sylbi 206 . . . 4 (𝑥𝑦𝑥𝑧)
109ad2antrl 760 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
11 noel 3878 . . . . . 6 ¬ 𝑥 ∈ ∅
12 noel 3878 . . . . . 6 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
1311, 122false 364 . . . . 5 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14 df-br 4584 . . . . 5 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1513, 14bitr4i 266 . . . 4 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
1615a1i 11 . . 3 (⊤ → (𝑥 ∈ ∅ ↔ 𝑥𝑥))
172, 7, 10, 16iserd 7655 . 2 (⊤ → ∅ Er ∅)
1817trud 1484 1 ∅ Er ∅
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ⊤wtru 1476   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  Rel wrel 5043   Er wer 7626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-er 7629 This theorem is referenced by: (None)
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