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Theorem 0erOLD 7729
Description: Obsolete proof of 0er 7728 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 5206 . . . 4 Rel ∅
21a1i 11 . . 3 (⊤ → Rel ∅)
3 df-br 4616 . . . . 5 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4 noel 3897 . . . . . 6 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
54pm2.21i 116 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
63, 5sylbi 207 . . . 4 (𝑥𝑦𝑦𝑥)
76adantl 482 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
84pm2.21i 116 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
93, 8sylbi 207 . . . 4 (𝑥𝑦𝑥𝑧)
109ad2antrl 763 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
11 noel 3897 . . . . . 6 ¬ 𝑥 ∈ ∅
12 noel 3897 . . . . . 6 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
1311, 122false 365 . . . . 5 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14 df-br 4616 . . . . 5 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1513, 14bitr4i 267 . . . 4 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
1615a1i 11 . . 3 (⊤ → (𝑥 ∈ ∅ ↔ 𝑥𝑥))
172, 7, 10, 16iserd 7716 . 2 (⊤ → ∅ Er ∅)
1817trud 1490 1 ∅ Er ∅
Colors of variables: wff setvar class
Syntax hints:  wb 196  wtru 1481  wcel 1987  c0 3893  cop 4156   class class class wbr 4615  Rel wrel 5081   Er wer 7687
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 4743  ax-nul 4751  ax-pr 4869
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 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-er 7690
This theorem is referenced by: (None)
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