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Theorem asymref 5250
 Description: Two ways of saying a relation is antisymmetric and reflexive. is the field of a relation by relfld 5395. (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 4213 . . . . . . . . . . 11
2 vex 2959 . . . . . . . . . . . 12
3 vex 2959 . . . . . . . . . . . 12
42, 3opeluu 4715 . . . . . . . . . . 11
51, 4sylbi 188 . . . . . . . . . 10
65simpld 446 . . . . . . . . 9
76adantr 452 . . . . . . . 8
87pm4.71ri 615 . . . . . . 7
98bibi1i 306 . . . . . 6
10 elin 3530 . . . . . . . 8
112, 3brcnv 5055 . . . . . . . . . 10
12 df-br 4213 . . . . . . . . . 10
1311, 12bitr3i 243 . . . . . . . . 9
141, 13anbi12i 679 . . . . . . . 8
1510, 14bitr4i 244 . . . . . . 7
163opelres 5151 . . . . . . . 8
17 df-br 4213 . . . . . . . . . 10
183ideq 5025 . . . . . . . . . 10
1917, 18bitr3i 243 . . . . . . . . 9
2019anbi2ci 678 . . . . . . . 8
2116, 20bitri 241 . . . . . . 7
2215, 21bibi12i 307 . . . . . 6
23 pm5.32 618 . . . . . 6
249, 22, 233bitr4i 269 . . . . 5
2524albii 1575 . . . 4
26 19.21v 1913 . . . 4
2725, 26bitri 241 . . 3
2827albii 1575 . 2
29 relcnv 5242 . . . 4
30 relin2 4993 . . . 4
3129, 30ax-mp 8 . . 3
32 relres 5174 . . 3
33 eqrel 4965 . . 3
3431, 32, 33mp2an 654 . 2
35 df-ral 2710 . 2
3628, 34, 353bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2705   cin 3319  cop 3817  cuni 4015   class class class wbr 4212   cid 4493  ccnv 4877   cres 4880   wrel 4883 This theorem is referenced by:  asymref2  5251  letsr  14672 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-res 4890
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