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Theorem intasym 4759
Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
intasym ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem intasym
StepHypRef Expression
1 relcnv 4753 . . 3 Rel 𝑅
2 relin2 4504 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 4474 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I )))
41, 2, 3mp2b 8 . 2 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ))
5 elin 3165 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
6 df-br 3806 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 vex 2613 . . . . . . . 8 𝑥 ∈ V
8 vex 2613 . . . . . . . 8 𝑦 ∈ V
97, 8brcnv 4566 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 df-br 3806 . . . . . . 7 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
119, 10bitr3i 184 . . . . . 6 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
126, 11anbi12i 448 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
135, 12bitr4i 185 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
14 df-br 3806 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
158ideq 4536 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1614, 15bitr3i 184 . . . 4 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1713, 16imbi12i 237 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
18172albii 1401 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
194, 18bitri 182 1 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wcel 1434  cin 2981  wss 2982  cop 3419   class class class wbr 3805   I cid 4071  ccnv 4390  Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399
This theorem is referenced by: (None)
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