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Theorem intasym 5025
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 5018 . . 3 Rel 𝑅
2 relin2 4757 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 4726 . . 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 3330 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
6 df-br 4016 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 vex 2752 . . . . . . . 8 𝑥 ∈ V
8 vex 2752 . . . . . . . 8 𝑦 ∈ V
97, 8brcnv 4822 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 df-br 4016 . . . . . . 7 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
119, 10bitr3i 186 . . . . . 6 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
126, 11anbi12i 460 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
135, 12bitr4i 187 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
14 df-br 4016 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
158ideq 4791 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1614, 15bitr3i 186 . . . 4 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1713, 16imbi12i 239 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
18172albii 1481 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
194, 18bitri 184 1 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1361  wcel 2158  cin 3140  wss 3141  cop 3607   class class class wbr 4015   I cid 4300  ccnv 4637  Rel wrel 4643
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646
This theorem is referenced by: (None)
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