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Theorem 0er 6051
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 4389 . . . 4 Rel ∅
21a1i 9 . . 3 ( ⊤ → Rel ∅)
3 df-br 3739 . . . . 5 (xy ↔ ⟨x, y ∅)
4 noel 3205 . . . . . 6 ¬ ⟨x, y
54pm2.21i 562 . . . . 5 (⟨x, y ∅ → yx)
63, 5sylbi 114 . . . 4 (xyyx)
76adantl 262 . . 3 (( ⊤ xy) → yx)
84pm2.21i 562 . . . . 5 (⟨x, y ∅ → xz)
93, 8sylbi 114 . . . 4 (xyxz)
109ad2antrl 463 . . 3 (( ⊤ (xy yz)) → xz)
11 noel 3205 . . . . . 6 ¬ x
12 noel 3205 . . . . . 6 ¬ ⟨x, x
1311, 122false 604 . . . . 5 (x ∅ ↔ ⟨x, x ∅)
14 df-br 3739 . . . . 5 (xx ↔ ⟨x, x ∅)
1513, 14bitr4i 176 . . . 4 (x ∅ ↔ xx)
1615a1i 9 . . 3 ( ⊤ → (x ∅ ↔ xx))
172, 7, 10, 16iserd 6043 . 2 ( ⊤ → ∅ Er ∅)
1817trud 1237 1 ∅ Er ∅
Colors of variables: wff set class
Syntax hints:  wb 98  wtru 1229   wcel 1374  c0 3201  cop 3353   class class class wbr 3738  Rel wrel 4277   Er wer 6014
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-er 6017
This theorem is referenced by: (None)
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