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Theorem intasym 5953
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 5945 . . 3 Rel 𝑅
2 relin2 5663 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 5634 . . 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 3924 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
6 df-br 5043 . . . . . 6 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 vex 3472 . . . . . . . 8 𝑥 ∈ V
8 vex 3472 . . . . . . . 8 𝑦 ∈ V
97, 8brcnv 5730 . . . . . . 7 (𝑥𝑅𝑦𝑦𝑅𝑥)
10 df-br 5043 . . . . . . 7 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
119, 10bitr3i 280 . . . . . 6 (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
126, 11anbi12i 629 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
135, 12bitr4i 281 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑥))
14 df-br 5043 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
158ideq 5700 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
1614, 15bitr3i 280 . . . 4 (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
1713, 16imbi12i 354 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
18172albii 1822 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
194, 18bitri 278 1 ((𝑅𝑅) ⊆ I ↔ ∀𝑥𝑦((𝑥𝑅𝑦𝑦𝑅𝑥) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wcel 2114  cin 3907  wss 3908  cop 4545   class class class wbr 5042   I cid 5436  ccnv 5531  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540
This theorem is referenced by: (None)
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