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Theorem intasym 6135
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 6122 . . 3 Rel 𝑅
2 relin2 5823 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 5792 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I )))
41, 2, 3mp2b 10 . 2 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ))
5 elin 3967 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
6 df-br 5144 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 vex 3484 . . . . . . . 8 𝑥 ∈ V
8 vex 3484 . . . . . . . 8 𝑦 ∈ V
97, 8brcnv 5893 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 df-br 5144 . . . . . . 7 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
119, 10bitr3i 277 . . . . . 6 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
126, 11anbi12i 628 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
135, 12bitr4i 278 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
14 df-br 5144 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
158ideq 5863 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1614, 15bitr3i 277 . . . 4 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1713, 16imbi12i 350 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
18172albii 1820 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
194, 18bitri 275 1 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  cin 3950  wss 3951  cop 4632   class class class wbr 5143   I cid 5577  ccnv 5684  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by: (None)
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