ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  0er GIF version

Theorem 0er 6568
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 4751 . . . 4 Rel ∅
21a1i 9 . . 3 (⊤ → Rel ∅)
3 df-br 4004 . . . . 5 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
4 noel 3426 . . . . . 6 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
54pm2.21i 646 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑦𝑥)
63, 5sylbi 121 . . . 4 (𝑥𝑦𝑦𝑥)
76adantl 277 . . 3 ((⊤ ∧ 𝑥𝑦) → 𝑦𝑥)
84pm2.21i 646 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ∅ → 𝑥𝑧)
93, 8sylbi 121 . . . 4 (𝑥𝑦𝑥𝑧)
109ad2antrl 490 . . 3 ((⊤ ∧ (𝑥𝑦𝑦𝑧)) → 𝑥𝑧)
11 noel 3426 . . . . . 6 ¬ 𝑥 ∈ ∅
12 noel 3426 . . . . . 6 ¬ ⟨𝑥, 𝑥⟩ ∈ ∅
1311, 122false 701 . . . . 5 (𝑥 ∈ ∅ ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
14 df-br 4004 . . . . 5 (𝑥𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ ∅)
1513, 14bitr4i 187 . . . 4 (𝑥 ∈ ∅ ↔ 𝑥𝑥)
1615a1i 9 . . 3 (⊤ → (𝑥 ∈ ∅ ↔ 𝑥𝑥))
172, 7, 10, 16iserd 6560 . 2 (⊤ → ∅ Er ∅)
1817mptru 1362 1 ∅ Er ∅
Colors of variables: wff set class
Syntax hints:  wb 105  wtru 1354  wcel 2148  c0 3422  cop 3595   class class class wbr 4003  Rel wrel 4631   Er wer 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-er 6534
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator