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Theorem 0er 6170
Description: The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)
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 4489 . . . 4 Rel ∅
21a1i 9 . . 3 (⊤ → Rel ∅)
3 df-br 3792 . . . . 5 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4 noel 3255 . . . . . 6 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
54pm2.21i 585 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
63, 5sylbi 118 . . . 4 (𝑥𝑦𝑦𝑥)
76adantl 266 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
84pm2.21i 585 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
93, 8sylbi 118 . . . 4 (𝑥𝑦𝑥𝑧)
109ad2antrl 467 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
11 noel 3255 . . . . . 6 ¬ 𝑥 ∈ ∅
12 noel 3255 . . . . . 6 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
1311, 122false 627 . . . . 5 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14 df-br 3792 . . . . 5 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1513, 14bitr4i 180 . . . 4 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
1615a1i 9 . . 3 (⊤ → (𝑥 ∈ ∅ ↔ 𝑥𝑥))
172, 7, 10, 16iserd 6162 . 2 (⊤ → ∅ Er ∅)
1817trud 1268 1 ∅ Er ∅
Colors of variables: wff set class
Syntax hints:  wb 102  wtru 1260  wcel 1409  c0 3251  cop 3405   class class class wbr 3791  Rel wrel 4377   Er wer 6133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-er 6136
This theorem is referenced by: (None)
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