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Theorem intasym 4625
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((x𝑅y y𝑅x) → x = y))
Distinct variable group:   x,y,𝑅

Proof of Theorem intasym
StepHypRef Expression
1 relcnv 4619 . . 3 Rel 𝑅
2 relin2 4372 . . 3 (Rel 𝑅 → Rel (𝑅𝑅))
3 ssrel 4344 . . 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 3095 . . . . 5 (⟨x, y (𝑅𝑅) ↔ (⟨x, y 𝑅 x, y 𝑅))
6 df-br 3729 . . . . . 6 (x𝑅y ↔ ⟨x, y 𝑅)
7 vex 2530 . . . . . . . 8 x V
8 vex 2530 . . . . . . . 8 y V
97, 8brcnv 4434 . . . . . . 7 (x𝑅yy𝑅x)
10 df-br 3729 . . . . . . 7 (x𝑅y ↔ ⟨x, y 𝑅)
119, 10bitr3i 175 . . . . . 6 (y𝑅x ↔ ⟨x, y 𝑅)
126, 11anbi12i 433 . . . . 5 ((x𝑅y y𝑅x) ↔ (⟨x, y 𝑅 x, y 𝑅))
135, 12bitr4i 176 . . . 4 (⟨x, y (𝑅𝑅) ↔ (x𝑅y y𝑅x))
14 df-br 3729 . . . . 5 (x I y ↔ ⟨x, y I )
158ideq 4404 . . . . 5 (x I yx = y)
1614, 15bitr3i 175 . . . 4 (⟨x, y I ↔ x = y)
1713, 16imbi12i 228 . . 3 ((⟨x, y (𝑅𝑅) → ⟨x, y I ) ↔ ((x𝑅y y𝑅x) → x = y))
18172albii 1334 . 2 (xy(⟨x, y (𝑅𝑅) → ⟨x, y I ) ↔ xy((x𝑅y y𝑅x) → x = y))
194, 18bitri 173 1 ((𝑅𝑅) ⊆ I ↔ xy((x𝑅y y𝑅x) → x = y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1222   wcel 1367  cin 2885  wss 2886  cop 3343   class class class wbr 3728   I cid 3989  ccnv 4260  Rel wrel 4266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269
This theorem is referenced by: (None)
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