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Theorem 0er 6076
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 4405 . . . 4 Rel ∅
21a1i 9 . . 3 ( ⊤ → Rel ∅)
3 df-br 3756 . . . . 5 (xy ↔ ⟨x, y ∅)
4 noel 3222 . . . . . 6 ¬ ⟨x, y
54pm2.21i 574 . . . . 5 (⟨x, y ∅ → yx)
63, 5sylbi 114 . . . 4 (xyyx)
76adantl 262 . . 3 (( ⊤ xy) → yx)
84pm2.21i 574 . . . . 5 (⟨x, y ∅ → xz)
93, 8sylbi 114 . . . 4 (xyxz)
109ad2antrl 459 . . 3 (( ⊤ (xy yz)) → xz)
11 noel 3222 . . . . . 6 ¬ x
12 noel 3222 . . . . . 6 ¬ ⟨x, x
1311, 122false 616 . . . . 5 (x ∅ ↔ ⟨x, x ∅)
14 df-br 3756 . . . . 5 (xx ↔ ⟨x, x ∅)
1513, 14bitr4i 176 . . . 4 (x ∅ ↔ xx)
1615a1i 9 . . 3 ( ⊤ → (x ∅ ↔ xx))
172, 7, 10, 16iserd 6068 . 2 ( ⊤ → ∅ Er ∅)
1817trud 1251 1 ∅ Er ∅
Colors of variables: wff set class
Syntax hints:  wb 98  wtru 1243   wcel 1390  c0 3218  cop 3370   class class class wbr 3755  Rel wrel 4293   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by: (None)
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