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Theorem asymref 5222
 Description: Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5368. (Contributed by NM, 6-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
asymref
Distinct variable group:   ,,

Proof of Theorem asymref
StepHypRef Expression
1 df-br 4396 . . . . . . . . . . 11
2 vex 3034 . . . . . . . . . . . 12
3 vex 3034 . . . . . . . . . . . 12
42, 3opeluu 4671 . . . . . . . . . . 11
51, 4sylbi 200 . . . . . . . . . 10
65simpld 466 . . . . . . . . 9
76adantr 472 . . . . . . . 8
87pm4.71ri 645 . . . . . . 7
98bibi1i 321 . . . . . 6
10 elin 3608 . . . . . . . 8
112, 3brcnv 5022 . . . . . . . . . 10
12 df-br 4396 . . . . . . . . . 10
1311, 12bitr3i 259 . . . . . . . . 9
141, 13anbi12i 711 . . . . . . . 8
1510, 14bitr4i 260 . . . . . . 7
163opelres 5116 . . . . . . . 8
17 df-br 4396 . . . . . . . . . 10
183ideq 4992 . . . . . . . . . 10
1917, 18bitr3i 259 . . . . . . . . 9
2019anbi2ci 710 . . . . . . . 8
2116, 20bitri 257 . . . . . . 7
2215, 21bibi12i 322 . . . . . 6
23 pm5.32 648 . . . . . 6
249, 22, 233bitr4i 285 . . . . 5
2524albii 1699 . . . 4
26 19.21v 1794 . . . 4
2725, 26bitri 257 . . 3
2827albii 1699 . 2
29 relcnv 5213 . . . 4
30 relin2 4957 . . . 4
3129, 30ax-mp 5 . . 3
32 relres 5138 . . 3
33 eqrel 4929 . . 3
3431, 32, 33mp2an 686 . 2
35 df-ral 2761 . 2
3628, 34, 353bitr4i 285 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450   wceq 1452   wcel 1904  wral 2756   cin 3389  cop 3965  cuni 4190   class class class wbr 4395   cid 4749  ccnv 4838   cres 4841   wrel 4844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-res 4851 This theorem is referenced by:  asymref2  5223  letsr  16551
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