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Theorem intasym 4829
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 4823 . . 3 Rel 𝑅
2 relin2 4569 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 4539 . . 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 3184 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
6 df-br 3852 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 vex 2623 . . . . . . . 8 𝑥 ∈ V
8 vex 2623 . . . . . . . 8 𝑦 ∈ V
97, 8brcnv 4632 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 df-br 3852 . . . . . . 7 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
119, 10bitr3i 185 . . . . . 6 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
126, 11anbi12i 449 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
135, 12bitr4i 186 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
14 df-br 3852 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
158ideq 4601 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1614, 15bitr3i 185 . . . 4 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1713, 16imbi12i 238 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
18172albii 1406 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
194, 18bitri 183 1 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1288  wcel 1439  cin 2999  wss 3000  cop 3453   class class class wbr 3851   I cid 4124  ccnv 4451  Rel wrel 4457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460
This theorem is referenced by: (None)
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